Thank you both.
I looked into our quarantine arrangements before posting, because the last thing I would want would be for a Yorkie pup (or adult for that matter) to have to go into quarantine. 
Guess what? the UK Government (or a tiny department of it) has caught up

 . There is a Pet Passport Scheme which, until I looked closely into it, I thought it was only for animal travel between European Union countries. But there is now quite an extensive list of countries that are included. The USA being one. The scheme allows for non-quarantine transfers subject to certain Vet checks and certification (probably part of the US routine in animal care, anyway).