Take your pick from one of these arguments:
1. There's no such thing as 'road tax'.
2. Vehicle excise duty is not ringfenced for roads, so all taxes contribute to roads.
3. Rates of VED are determined by pollution, so you don't have to pay it for a bike because they don't have emissions.
4. Not having to pay is arguably a tax break to encourage people to cycle rather than drive. If you don't like paying it, get a bike.
If you lived in Tower Hamlets, where it's congested to death as it is and now made worse because of the increasing number of cycle lanes (not to mention the road closures to build cycle lanes) you'd be making not so well thought out arguments too.