Talk to the airline carriers about how to handle this States side... you can ask each airline for quotes along the way and what their policies are, because each one has different policies.
For example, British Airways World Cargo required me to drop off my two cats in Chicago before noon on the day of their international flight-- which was impossible for me to do coming from St. Louis and they weren't willing to accept them unless *I* or an approved courier took them there. But Continental gave me better rates and could start them out from St. Louis without a problem (though they had to fly down to Houston and then Newark before heading out to Manchester, I'll get to that). Also, most rates are flat rates so no matter where they have to ping them around within the US, you shouldn't get charged more.
The 'approved routes' only applies to the final departing US city to the first arriving UK city.
I didn't accompany my cats for the trip-- they went with cargo all of the way through, which was part of the reason they got flown from St. Louis to Houston first-- because it was one of the few routes that had a plane with a sealed cargo bay to accommodate them. So, I didn't have to handle their carriers at all. I imagine that's the same for any pet owner even if they are accompanying their pets on the same flight for connecting flights, with exception to seeing eye dogs.
Are you on the DEFRA Pets Scheme (got their 'Pet Passport')? Did you get their US Health Certification? Or are you importing them into quarantine?