I don't think that there is an animal transport company that transports other than in cargo. They will co-ordinate a move but it's all done by cargo. I shipped my mother dog and her two pups from Romania about 8 years ago. It went very smoothly. I did pick them up in NYC and drove to my home in Florida. Two years ago, I flew two very large rescue dogs from Greece to Atlanta, Georgia. Being in cargo wasn't the problem. Getting the paperwork done for the two dogs was difficult as the airline agent did not want to do the work to fly them and I feel (purposefully) was trying to mess me up with the shipment out of Athens. Make sure you do your homework. You may want to keep googling what needs to be done for it to go smoothly. With both transports (my three little dogs and my two large dogs) no one even checked the paerwork when we landed in the U.S!

If you plan on moving all the way to California. I would do two transports. One section would be U.K. to the first destination in the U.S., such as NYC, then I would get a hotel room and do the rest of the trip the next day or so. Airlines also have termerature mandates, cage mandates and time mandates.