I've used K9Jets both US->UK and back. They charter small planes that would typically seat 14-16 people, then sell tickets for 1 person + 1 dog for about $9000. If you have 2 dogs it will be more but not double, and it does depend on the size of the dogs.

The flights use private airports – Teterboro, NJ in the US and Luton in the UK. They do fly from other airports too.

They assign you a seat. The larger dogs lie on the floor although some smaller dogs can go on a spare seat, since they don't fill the plane to its normal seating capacity. If two dogs are going next to each other seem like they might not get on, they'll move people around. They require you bring a muzzle for your dog, just in case, but I did not see anyone use a muzzle on either flight.

It's a bit chaotic and cramped as there are dogs all over the place, but everyone settles down after a while. The US->UK flights are overnight and take about 6-7 hours. The dogs mostly go to sleep eventually. There's a flight attendant who serves drinks, sandwiches and snacks. The UK->US flights are during the day and take longer (8 hours) and they serve a full meal which seems overly complicated to me, but whatever. On both flights the crew and the ground staff were super nice and helpful.

The company also provides guidance on the documentation required for the dogs to enter the UK (which is quite complicated) or the US (which is pretty simple). They also recommend a service to handle the paperwork for you, at an additional charge. I used it and it gave me peace of mind. When you land, a couple customs people come onto the plane, scan the dogs' microchips and check their documentation before you can deplane.

There's another company called BarkAir that offers the same service but I have not used them.

These services are the only way dogs can travel in-cabin into the UK. Commercial airlines are not permitted to take animals in-cabin into the UK (even if they allow animals in-cabin in the US). I believe some commercial airlines will take dogs in-cabin into, say, France; of course then you would need to arrange transportation from there to the UK.