TL;DR
The brief answers are 1) yes, there were secret documents stolen by the Germans, and 2) no, the English radio did not mislead the British into the capture of men, equipment, and such documents. The story in Hitler's letter includes some real elements, mixed with falseness, exaggerations and misunderstandings, and perhaps (impossible to know) some lies.
Long answer
Fascinating! That is my conclusion from the analysis I carried out, motivated by @Schwern's post. I searched through a lot of books and even found real German documentary about the period. However the best source by far was a book written by Lieutenant-Colonel Dudley Clarke called Seven Assignments published in 1948. Clarke was a British officer, expert in military deception, who participated in several WWII campaigns. One of them was in Norway where he actually traveled as personal companion of Brigadier Morgan! Hence the relevance of his writings for the issue at hand. Other good sources are the reports by L.G. Massy, Commander-in-Chief of the North Western Expeditionary Force, sent to the Secretary of State for War, the 13th of May 1940, documented in Greham and Race (2015).
Before the start of the Allied invasion to Norway on the 13th of April of 1940, Clarke was sent up by the War Office in London to Rosyth in Scotland with "£10,000 in notes, ... [and] eight mailbags of maps and secret pamphlets." (p.81) He had to deliver these to Brigadier Morgan. Once his delivery was finished, he decided to accompany Morgan in his trip to Norway because returning to London "would be only the prelude to a further period of inactivity" (p.87). Once Morgan agreed to this, Clarke sent a telegram to the War Office in London to "inform" (rather than ask permission) them of his decision. So, in Part II of his book, Clarke recounts all his adventures with Morgan in Norway from the embarking until the 25th of April when Clarke returned to London.
So, let's approach the "real" (or British) version of events. The contrast with Hitler's story should be made clear then.
First, a mention about the "secret documents". As said, Clarke was sent up to Scotland with money, maps and secret documents. Clarke says the mailbags and documents were uploaded onto one of the ships. While they were waiting to depart, many activities were performed in the ships. He recounts:
"I tried to help the Brigade Intelligence Officer, who was emptying the mailbags from London and separating a mass of maps and Intelligence Summaries into smaller parcels for the various detachments." (p.92)
So, it seems there were some secret documents ("Intelligence Summaries") in the ship, together with some maps. Interestingly, the destination of Morgan's mission was not clear by then. Whereas the last communication Morgan received stated they were to sail to Namsos (in Norway), en route he received an update. Clarke again:
"During the afternoon he [Morgan] held a conference for his staff, at which we learned that the objective was not to be Namsos after all. Instead SICKLE FORCE was to make an entirely fresh landing at Aandalsnes, some 300 kilometers further south. ... my own reaction to this new development centered around the hundreds of maps which have been carted along so diligently in the mailbags all the way from the War Office [in London]. None of them, I knew, covered Aandalsnes at all." (p.94-5)
The trip apparently was not very pleasant. Plevy's book says:
"After a rough crossing of some forty hours, with much sea-sickness among the troops caused by the violent zigzagging to the ships to counter U-boat attacks, the reinforcements ... were landed at Molde and Aandalsnes late on 18 April." (p.147)
Hitler wrote that the British were still suffering from seasickness when they arrived in Lillehammer. But as this happened the 20th, 3 days after landing, is this likely? Who knows.
Clarke notes that upon arrival to Aandalsnes, there was some information about the state of the frontlines:
"The Norwegian Commander-in-Chief, General Ruge, we learned, had his G.H.Q. at Lillehammer, 150 miles up the railway, with the remnant of his army holding the Germans astride Lake Mjosa." (p.99)
Regarding physical communication routes, unlike Hitler's story, German aerial strikes had not destroyed the railways. Regarding the Gudbrandsdal valley, connecting Aandalsnes and Lillehammer, "which is seldom more than a mile wide and in places narrows to a few hundred yards" (Greham and Race, 2015, p.10), Messy says:
"It would be difficult to imagine a Line of Communications more exposed to air attack, to which it was continually subjected during the hours of daylight by means of heavy bombing and machine gun fire: and there were no means of protecting it nor of repairing the damage done to the roads and railways: for this latter work reliance had to be placed entirely on the Norwegians, who did their best with very limited resources. The key point of Dombaas was completely destroyed by bombing and Otta almost completely so. Large craters on the road made motor transport movement increasingly difficult; it was singularly fortunate that the railway was not more seriously damaged." (p.10-1).
So maybe the heavy bombardment gave the Germans the impression that railways had been completely destroyed. But no, trains were still available. Clarke himself says:
"At least the railway up to Dombaas Junction was still open and working." (p.99)
As it happens, it was open all the way to Lillehammer. Furthermore, the British did face some opposition when moving between Aandalsnes and Lillehammer, as German parachute troops were constantly being deployed, and taking of civilian hostages in the villages did occur (see Clarke's account of events, p.99-102).
Upon arriving to Aandalsnes, Morgan immediately sent an expedition to Dombaas, to which he and Clarke joined, not fully knowing if it was under Norwegian control. There he would asses the situation. Later in the day, a message from Lillehammer arrived by telephone (again, suggesting not all lines of communication were cut-off, and a reliance on any potential "English radio" message is thus non-sense), requesting Morgan to wait for the Norwegian generals at Dombaas. Once they met, the generals asked Morgan to help them in the area south of Lillehammer where they were resisting. After agreeing, the British troops went to Lillehammer. Importantly, the British did not "march" toward Lillehammer (as Hitler suggests), but got there by train. Massy again:
"The force [148th Infantry Brigade] was moved by train to the Lillehammer area during 19th and 20th April" (p.11)
Now, the British did not stay there, but moved further south where the front was. Harry Plevy (2017) says:
"By the evening of 20th April, both of his [Morgan] battalions would be spread out on a wide front, some ten miles south, south-west and south-east of Lillehammer, ..." (p.154)
Unfortunately for the British and Norwegians, things did not go well. Plevy writes that on 21st of April:
"[a]t about 6 p.m., General Hvinden-Haug ordered a general withdrawal north of Lillehammer, with the British units covering the withdrawal." (p.156)
Regarding the loss of men, Greham and Race (2015) say:
"During this withdrawal a party of 5 officers and 50 men, 1/5 Leicesters, was cut off and lost." (p.11)
Similarly, from Kiszely (2017):
"As the Leicesters' War Diary recorded succinctly, 'The Germans caught up the battalion, and a great many of them were cut off.' A number of these groups were taken prisoner, including on of five officers and fifty men from the Leicesters. The Germans were soon in possession of Lillehammer, capturing British ammunition and stores." (p.187)
This is an indication that Lillehammer was not captured by surprise as in Hitlers account.
Now at the time of the attack Clarke was in the British HQ in Oyer, around 10 miles north of Lillehammer, with Morgan (meaning Morgan was not captured in Lillehammer!). Aware of the German reinforcements in Oslo and the fall of the front, he feared "for the safety of the secret papers and cipher books from the Legations." (p.121). Later on, whilst he was still there, the Germans attacked them. I quote in extenso this remarkable event:
It was one of the sentries at the back who presently called our attention to the mountain top, where the distant figures of skiers were now appearing. ... As we strained through field-glasses to distinguish uniforms, any remaining doubts as to whether they were Germans were soon dispelled. There was only one thing left to do then; and that was to get out quickly... Precious moments were spent in trying first to destroy the sackloads of Intelligence Summaries and secret documents, while the Field Cashier dragged out one of his treasure chests [with the £10,000?] in the direction of Morgan's car. The next few minutes were a nightmare. The wooden inn contained a single iron stove with one small opening, so that it was necessary to tear up the voluminous papers, after first shaking them into the stove slowly enough to prevent the fire being blanketed out altogether. While this well-nigh hopeless task was in progress Morgan returned from the road to say that the civilian lorries ... had disappeared, and that his own car alone remained. Meanwhile ... the figures of the skiers were looming uncomfortably large. ... On that Morgan ordered the place to be abandoned. ... One final attempt was made upon the papers by starting a bonfire of them on the floor in the hopes of setting the house alight, and my last view was of the Brigadier Major [Morgan] feeding the flames with packets of crisp £1 notes. Then I clasped my own precious bundle and joined Morgan outside ... while a satisfactory cloud of smoke followed from the doorway of the inn. ... in the end we got clear and sped away round a corner of the road which hid us just in time. It had been a near thing, so near we learned afterwards that the Germans were unfortunately able to get there in time to put out the fire and recover some of the papers. (p.126-7)
This tale suggests that the documents might not have been captured in Lillehammer but further north. In any case, later on Clarke was present in the evacuations of central Norway just as Morgan was (suggesting again the latter was not captured in Lillehammer).
The story of the capture of Lillehammer, the British POW and the documents can also be seen in a Nazi propaganda film called Kampf um Norwegen - Feldzug 1940, made in 1940 (but "found" only in 2005), showing the German campaign in Denmark and Norway. At 48:10 the narrator says that Lillehammer was taken by surprise early in the morning the next day (i.e. 21th of April). In the video, you can see Germans walking around Lillehammer. At 48:25 a group of English POW are shown, which according to the narrator represents the first time English and German troops have been front to front. If this is true, then it is clear these POW are those that could not escape Lillehammer, as these was in effect the first large capture of British soldiers in Norway (the similar weather in all the scenes might further support these were English POW from Lillehammer). If you run this bit at 0.25 speed, you can actually count the soldiers: 9 pairs of 2 (although the last one seems to be alone); 17 or 18 soldiers (not sure how one would recognize officers there). If by any chance the soldiers were divided into two groups, that was one of the groups. The other could have passed before. In any case, in that scene the narrator says:
"British documents that fell into our hands contained orders for the planned occupation of Norway on April 8."
Did these documents contain details about Operation Wilfred? What else did they contain?
In summary, although Hitler's letter does describe, broadly speaking, events that did happen (i.e. German air bombing, the arrival of 148th battalion to Lillehammer, and the capture of British soldiers and secret documents) the details associated with them are all wrong. The detail about the English radio is also not true. Even if the radio could have reached those positions, it is clear that the actions of Morgan and the Norwegian army was not based on any such report.
