Sounds like a good idea, but wouldn't we have to extend it further?
As people have said, cyclists who also have cars will have already paid road tax, but are using their cars less. Car owners who also have bikes will now be paying bike tax, whether or not they're using them. We'd also have to decide whether you pay to license each bike / car, or whether you pay some kind of 'right to cycle / drive' tax.
Most logically, since most people have both a bike and a car, we ought to charge both the car tax and the bike tax per mile travelled, rather than as a flat payment. Probably trackers would need to be fitted to the cars and the bikes in order to track usage.
Now, people would be paying explicitly to use their roads, their cycle paths, but not their use of pavements. Presumably, therefore, we should also charge an explicit pavement tax. People who pay high levels of road / bike path tax would presumably not be using pavements, and so should pay less pavement tax.
Pavement tax could perhaps be rebated for areas of the country where there are few pavements.
Cars / bikes / people from outside the UK could pay some kind of flat fee entitling them to a certain amount of road / cycle path / pavement usage during their stay in the UK.