The Thompson was dropped before Korea, and supplanted by the M3 grease gun which was in service (issued to tankers) until the 80's. many Thompsons ended up in the Phillipenes and other US allies hands post WW2.
Select fire M-16's and Car/XM177 replaced the .45cal submachine gun in main line units sometime in the Vietnam era, though the M-3 was issued/given away to Montagnards and other RF/PF forces in Vietnam and saw use with Spec ops.
Before the HK was an 'official' weapon of US forces it was a 'clean' weapon, like the Swedish K that could be 'denied' having US origins. It was also issued to Iranian forces in the Shah era, and I'm pretty sure one of the first deployments of the MP-5 was to Delta Force for the failed hostage rescue attempt.
There was a drawing of an engraved HK MP5 with 6 pounds of ivory made as a presentation weapon for the shah in Gun Digest in the 80's.
.45 cal ammo was never a NATO standard, the switch to a 9mm SMG came before the swith to a 9mm sidearm, teh 9mm SMG is still only issued in limited numbers.
(On a side note, I know a guy who served as a Bradly commander in Desert Storm that was issed a 1911a1. He said it was the most worn out handgun he'd ever handled but it went bang every time. The switch to 9mm handguns had a lot to do with the supply of .45's being old and worn out, as well as the switch to NATO standard ammo.)
The thompson is a HEAVY weapon that's hard to machine, whereas the MP5 is light, using stamped parts. The later M1A1 Thompson had rudimentary sights, while the 1928 had an overly complicated Lyman sight (given what the weapon was designed for). The M1a1 is a simplified 1928 design, the HK is a design derived from the best of German WW2 weapons and US Vietnam era weapons ie stampings, in-line recoil, plastic parts, alloy parts etc.