The simplest option would be to instruct a pet shipping agent in the US and customs clearance agent in the U.K. There is a lot of red tap to navigate and the slightest error could cost you hundreds if not thousands of pounds in fees if the dog arrives non-compliant in the U.K. There is also the question of airline breed restrictions, temperature embargos, and routing to consider.
If they are genuine pets (non-commercial) and you can prove it, there is little in terms of VAT/duty to worry about. If you’re importing a dog you’ve recently purchased, there will most likely be VAT to pay at the going rate. Something to bear in mind.
You can select an agent from here
Search Online for a Pet Shipper or pm me with details of the relocation. There are too many variables to list specifics without more info.
Using a lab as a guide, and a very rough guide at that, you could be looking at £2000 - £3000 for the US side and £250-£500 the U.K. side assuming the paperwork is in order. The bulk of the export cost is freight and airline handling.