Well, I've done some research, and reached some conclusions.
If Italy had fallen to the Allies in autumn 1943, this could have had a drastic effect on the war. Bulgaria and Rumania might well have changed sides earlier, and the resulting loss of raw materials from the Balkans and Turkey would have badly damaged German production by mid-1944, as stocks were used up. The war probably would have ended significantly earlier, but the lengthy defence of the Winter Line by the Germans in 1943-44, including the battles of Monte Cassino, prevented that. So the belief of the Allied troops in Italy that victory there would bring the end of the war closer was true at first, but they kept on believing it after it stopped being so.
Ellis, in Cassino: Flawed Victory, reckons Truscott was upset by the diversion of VI Corps, because the situation was looking good at the time for blocking the road, contrary to what Clark claimed afterwards. He also says that British 8th Army pursued German 10th Army rather slowly, due to poor staff work causing traffic jams on the roads, and a general lack of keenness. I suspect they might have concentrated on the task rather better if they hadn't known that Clark had abandoned the blocking move in favour of a triumph in Rome. Some of Alexander's decisions seem to have been prompted by his annoyance by Clark's actions. They might thus have been different had Clark allowed the road to be blocked. The British certainly believed that a successful blocking of 10th Army's retreat would have brought a rapid end to the war in Italy.
Taking Northern Italy in August-September 1944 has several effects:
Germany no longer has to supply coal to keep the industries running, and the Allies probably can't supply it soon enough and fast enough to get production moving significantly before the war ends, although they'll try. Historically, the coal supply stopped in early 1945 when the German transport network collapsed due to the bombing under the Transportation Plan (see below). The lack of need to supply Italy might delay that a day or two, but not much more; the network collapsed thoroughly.
This means Germany looses about 15% of its arms production for September-December 1944, because that depended on Italian production and sub-contracting.
Germany also loses some manufacturing of consumer goods and food production in Italy. This increases the suffering of the German civilian population, especially those that need re-housing after bombing, but probably doesn't speed the German military collapse noticeably.
The Allied tactical air force in Italy is able to attack the south of the Reich from November 1944, which they didn't manage at all historically. It makes little difference to the strategic air force in Italy, although they'll be able to carry somewhat greater loads or go further north.
The Romanians historically switched sides in August 1944, and the Bulgarians in September 1944. The Yugoslav partisans can disrupt the railways thoroughly by then, so no materials are coming out of the Balkans; they were important in 1943-44, but by this timescale, they were ending anyway.
Turkey and Spain may stop supplying materials to Germany sooner than February and May 1945 respectively. Swiss economic co-operation with Germany ends in August 1944 anyway, when the Allies reach the Franco-Swiss border. The Swiss supplied dairy products, optical and mechanical parts, machine tools, railway engines, munitions, explosives and electricity.
Going East from the Po valley allows the Allies to cut off German Army Groups E and F in Yugoslavia, depriving them of supplies from the Reich and meaning that Greece and Yugoslavia get liberated sooner.
Meanwhile the Allied troops from Dragoon that stayed in the French Alps guarding against attacks through the passes from Italy into southern France aren't needed any more. They can go north and help with Germany. Their identity is obscure, because this was a front where not much of significance happened, but they seem to have been French mountain infantry, who were facing the German 34th Infantry Division and 5th Mountain Division, plus the 2nd Division "Littorio" of the RSI army and other RSI units. I've started another question about that point.
If the Allies start pushing north into Austria in the spring, the Alpine Redoubt looks a lot less credible, and the arms production in Vienna has been bombed by tactical air forces all winter. Austria may be entirely conquered by the Western Allies, rather than being shared with the Soviets, depending on how much fight the Germans put up in Hungary.
Kesselring sent the complete Sicilian order of battle to OKW via Enigma shortly before the start of Husky.
Hitler seems to have expected an invasion of the Balkans until Husky was well underway, partly because he thought the Allies had much more strength in the Med than was actually the case, thanks to strategic deception.
Canaris' assurance that the Allies weren't planning any landings shortly before Anzio happened was one of the things that cost him his job.
The very limited Luftwaffe presence in Italy was transferred to Germany in September 1943, as it was needed there to fight the Anglo-American air forces.
The German Side of the Hill, PhD thesis, 1999, Timothy D Saxton.
Germany and the Second World War, volume V/IIA, the German official history.
The Collapse of the German War Economy, 1944-45, Alfred C Mierzejewski.
Cassino, the Hollow Victory, John Ellis.
Impossible Victory, Brian Harpur.
Appendix: How bombing finally managed to disrupt the German economy
The Combined Bomber Offensive was intended, by its British and American commanders, to win the war against Germany all by itself. The slight problem was that they didn't really know how to do it, and it was quite some time before they found out.
The Oil Campaign of bombing killed off the synthetic oil plants: the key was to monitor the repairs via photo-reece, and bomb again before production re-started. That ended the Luftwaffe and de-mechanised the Heer.
Meanwhile, the Transport Plan totally disrupted German industry, which was almost entirely coal-powered and relied on coal-fired steam railways to deliver coal and coke, to move iron ore to the iron and steel plants in the coal fields, and to move steel to the armaments plants which were mostly in different parts of Germany.
The railways were very efficient, if undamaged, but were working very close to their absolute capacity. Sustained bombing of the marshalling yards built up snarls of wagons that were unusable because they were in the wrong place and couldn't be got to the right place. The railways' telephone and telegraph systems ran alongside the lines, and were also wrecked by bombing, which meant that re-organisation became impossible, in the face of continued damage. And everything fell apart in Jan/Feb 1945.
The list of raids on that page is wildly incomplete; Wikipedia's coverage of the Transport Plan is somewhat lacking in general.