Traditionally, the european divisions of Ford and GM operated almost independantly of their American counterparts and so their cars often had little in common with their stateside offerings.

The other problem is that American engines tend to be bigger than European ones and their gearboxes are beefed up accordingly. A gearbox designed for a 3.5 V6 isn't going to make a good partner for a 1.0 econobox.

Things are starting to change with the "world platform" concept where a common design is used in all markets to cut costs.

US autoboxes can be pretty good. The ubiquitous GM TH400 found in most of GM's larger family cars was used by Rolls Royce. Don't tell RR fans. They like to think that every component in their car is handcrafted from sold silver by a wizened old craftsman with a woodbine stuck to his lower lip....

These days Chevrolets are rebadged Daewoos, at least the ones that would fit our roads are.

Bigger American cars use longitudinal engines and rear drive so their transmissions would be in low demand here.

Smaller American cars with transverse engines and front drive use Japanese transmissions - GM using Aisin (Toyota) - Ford used to use Jatco (Nissan) transmissions but now use Aisin.

There's also a perception that US-built cars have lower build quality than Far East or European-built cars.

Well first off the vast majority of Fords sold in the UK are made in Belgium. There's very little American about Chevrolets you buy over here as they're re-badged Daewoos. Chryslers in the UK are mostly made on a Fiat platform and built in Poland.