On nearly all boilers the pilot flame will only light if the fan is running and has been recognised by the air pressure switch inside the enclosure - there is not much you can reasonably do if you want a quick fix.
I get the impression that if you call British Gas they will want to sell you a new boiler and flush your radiators too - they tend to up-sell.
Ask a neighbour for a recommendation
Small is good...
September 2015