The ambassador most likely left Poland for Romania by about noon on Sept. 17, 1939, within a few hours of the Soviet invasion that morning, having already left Warsaw for Kuty in preparation for just such action.
From the House of Commons Hansard records for Sept. 20, 1939 - Columns 976-7 (my emphasis):
The Prime Minister (Mr. Chamberlain)
... On the morning of 17th September Russian troops crossed the Polish frontier at points along its whole length and advanced into Poland.
I cannot say that the action of the Soviet Government was unexpected. For some time past Soviet troops have been mobilised and concentrated on the western frontiers of the Soviet Union, and statements have appeared in the Soviet Press and wireless referring to the position of White Russians and Ukrainians in Poland, which bore the interpretation that the Soviet Government were preparing for intervention.
His Majesty's Ambassador to Poland, who was established in the Polish town of Kuty, near the Rumanian frontier, was advised by the Polish Government to leave Poland as soon as Russian troops crossed the frontier, and he is now in Rumania with his staff. I would like to say a word of sympathy with Sir Howard Kennard and the members of his staff, as well as His Majesty's Consular Officers in Poland. They have had to suffer such an ordeal of anxiety, fatigue and danger as seldom falls to the lot of members of their Services, but I need hardly say that they have carried out their duties with the courage, efficiency and disregard of personal considerations which we should expect of them.
The identical statement (italicized paragprah above) above was made to the House of Lords (Column 1082) an hour or so later by Earl Stanhope, Lord President of the Council. (there is a typographic error in the time, but it indicates some time after 4 p.m.
The previous sittings of the House being on Sept. 13, this relocation by the Ambassador could have happened on any of those days - however the Soviet Army didn't invade Poland until Sept. 17. So we have a time window of about 3 days, from whenever on Sept. 17 word of the Russian invasion reached Kuty until the morning of Sept. 20, for the relocation to Romania to have happened.
Given the clear expectation that the Soviets would invade, and the foresight to relocate so close to the Romanian border, I expect that the ambassador was in Romania by about noon on Sept. 17, 1939.
Kuty is located about mid-way between Kolomyja and the Romanian border, shown here by a small circled blue x, 20 to 30 km from the border: