I dont know about UK but in Sweden it works like this:
the company rent the plane from the owner.
The owner sends a invoice which includes the amount of fuel used (important), The company pays the invoice to the owner, you send a paper to the tax office reclaiming the fueltax on the fuel used for the businesstrip (it is only fuel used for leisure flights that you need to pay taxes for) If the owner has a VAT number the company can withdraw the VAT as well...
