Barret,
For you in North Somerset, you will come under the Welfare of Animals (Transport) (Engalnd) Order 2006, which implements EU 1/2005 - a very large piece of legislation.
This permits transport in containers, trailers or areas of a rigid vehicle that are suitable.
Two issues you will face.
The first is the suitabiility of the area they will be in. The regs state that no person shall transport animals or cause animals to be transported in a way likely to cause injury or undue suffering to them. They should be designed, constructed, maintained and operated so as to avoid injury and suffering and ensure the safety of the animals.
The second major issue you will face is loading and unloading. If you lift an animal, this must be done without risk of injury to the animal and whilst you can humanly lift a piglet, a slaughter weight would be impossible! The regs would expect you to have ramps (or a lift!) and for the ramps to have side protections so that the animal can walk out with a ramp angle of less than 36.4%!
Various other parts of the legislation might be infringed depending on how exactly you are transporting.
Additionally a slaughter house can reject any animal if they are not happy with transport - this is not part of the regs, just the ability for anyone running a business to put conditions on their trade with you.
Overall whilst an improvised solution can be made to satisfy regs, most won't!