UK fuel prices are not too cheap, in fact they are artificially high. The price of public transport, however, is too high to encourage use. If it were cheaper it would be used more often by those without cars and therefore there would be more viable routes and more incentive to use public transport.
Fuel prices, even with recent price rises are as cheap, relatively, as they were before the 70s oil price crisis and we see why people think nothing of jumping in the car, often multiple times in a day for relatively short journeys. Add to that, cars are bigger, heavier, and less fuel efficient than they were even a few years ago.
Bus and rail fares are far too expensive - a result of privatisation and the hundreds of millions taken out of the system in payments to shareholders who have made huge profits out of what were once public assets sold at knock-down prices.
If we want to discourage driving, we need the cost of fuel to increase significantly, as well as finding other ways of making driving less attractive. Certainly making public transport cheaper and more accessible would be a big help. Personally I'd be all in favour of renationalising the railways and for local authorities to take over the running of the buses again (Edinburgh is one of a few places still with a successful council-owned bus service).