What is the best bike to do a road/trail coast to coast, for around £500?

Get a touring bike it will be the best all around bike to get. They will handle smooth dirt and gravel bike paths are are more comfortable then the average road bike.

I have a cannondale F5 and it rolls great on the road with either some hard dirt tires like Geax evolution, or road tires like Geax roadster. I've done a few centuries, just lockout the fork and inflate the tires to max pressure. Not quite as efficient as a road bike, but as comfortable as a hybrid, handles trails with ease with road tires, and has rear pannier mounts. My son has a Trek 4300 that I would say is about the same. Look locally on Craigslist or local bike recyclers, ebay can be a bear in purchasing bikes.

you need a touring bike but you won;t find one for 500 though quid or dollars used is ok if you know what you are doing oh charity ride maybe you don;t need touring since you prob won;t be carrying any significant gear and a car will be following for breakdowns yeah get a hybrid any of the brands be sure it fits and you can ride 40 miles without butt meltdown, first anything is fine get something with mounts for fenders, panniers and racks touring would have all that hybrid maybe maybe not wle

I would not go second hand for that long of a trip at all. Try on ebay and for a new bike. I've only done local travel, but I would recommend a hybrid commuter bike, like these: http://www.2wheelbikes.com/hybrid-commuter.html. They're comfortable.