I used Eric Gillie to transport my two from Surrey to Scotland. Found the cost very reasonable and the horses arrived safe and sound. Boxes have partitions from the roof to the floor and are air conditioned so very secure and comfortable.
I can't remember the number but if you do a search on the net for eric gillie (Kelso) you should get it.
He transports racehorses and has a fleet of boxes over the country so should be able to help you.
I would expect to pay around £250 for that sort of trip though on an individual load on a specific day slightly cheaper for a shared but then you won't get a direct trip so endless miles and if your horse is going to newmarket sounds like you may be visiting the vets so go individual.
If you are going to the vets your insurance company may even pay transport costs (Kbis, nfu and SEIB all pay transport costs with a receipt)