Leigh Day will not 'cover the legal expenses'. You will have a Conditional Fee Agreement with Leigh Day the same as you will with Kevin O'Sullivan.
BC, CyclingUK, etc. try to suggest that if you use their linked solicitors then somehow the legal fees are covered - they're not. You will end up with a Conditional Fee Agreement the same as with any other legal firm/solicitor and they will recover their fees from the other side (if you win).
What BC et al might cover is any insurance premium (although that too will be recovered from the other side if you win).
Personally, from others' experiences I wouldn't go with Leigh Day. I'd look at Slater & Gordon.
Thanks for the clarification - i have subsequently realised that this is the case, I misunderstood the nature of the cover from BC.