Actually one Avro York, the rest had Melrins.
The Hercules was sketched on paper for a number of single engine aircraft in the late 30s. Manufacturers often offered several different engines (or the air ministry requested possible alternatives).
Unfortunately the Hercules suffered from sort of a triple whammy at a critical time. The difficulty in mass producing it has been mentioned many times and that got sorted out just in the nick of time. However power development seem to have stalled out about the same time. The Merlin was having no such difficulties and became the engine of choice in a big way. Yes 2000hp engines were planned but something was needed in the meantime, the 2000hp engines being several years away from production.
Cooling at higher powers became a problem with much research put into new heads with more fin area.
3rd hit. The rapidly improving fuels allowed much higher boost pressures to be used in some engines but the Hercules was not able to take advantage of the full potential, inpart due the already mentioned heat and in part because the sleeves would not withstand the increased cylinder pressure as the ports began to open on the exhaust stroke/cycle. The unsupported edges of the openings in the sleeves would crack or distort at high pressures. More work solved this problem (somewhat) but even the 100 series engines only ran around 8-9lbs max boost.
Post war use (or low use) of the Sleeve valve engines may not prove much. Britain and the British Commonwealth had a huge cash problem (they didn't have any) and imports of much of anything was severely restricted. Some of our British members may have stories of how long rationing of certain items lasted (into the 50s I believe???) in any case there was no money to pay for American engines in 1946-48 or later. It was British engines only (or commonwealth engines) to help with this balance of trade/cash problem.
As noted some of the last British sleeve valves could sometimes post tremendous times between overhauls but airline use is pretty hard nosed when it comes to cost.
American engines may have been cheaper for the rest of the world and parts easier to get. Commercial R-2800 engines had pretty good overhaul life. A plane sitting at an airport in Africa or asia waiting for engine parts wasn't making any money so replacement engines and parts were part of the overall calculation.
Unfortunately for the British their share of the market was too small to justify very many non British commonwealth airlines from signing up.
Cheap C-47s, C-46s and C-54s pretty much killed the airliner market for France and Italy except for a few prototypes or flag waving small production runs. Same of any French or Italian commercial piston engines. Immediately post war there was a flurry of activity that lend to not much, French were even trying to sell brand new Jumo 213s since the Germans had left behind a production line. It didn't go far. The French powered about 2 dozen twin engine flying boats with them. How much of this was to provide employment or try to use french resources while recovering from WW II I don't know.
It was a strange time economically and lots of decisions were based on these trade balance issues and not on which was the best engine or aircraft (unless something was a real turd).
P & W post war production of R-2800s helps explain the longevity/dominance of American engine engines in the commercial market.
1946-472
1947-1172
1948-956
1949-478
1950-546
1951-817
1952-1056
1953-2522
1954-2086
1955-764
1956-583
1957-691
1958-384
1959-3
1960-7
over 12,500 R-2800 engines.
Granted a lot of these are military engines but there was a very large market for engines, used engines, spare parts and overhaul services.
The sheer quantity ensured a longer working life than most other engines, regardless of individual merit.