Fuel duty is also collected by central government, and none of it goes to the London mayor either.
The only taxes collected and managed by London are the Congestion charge and the ULEZ charge. Everything else is beyond the reach of the London mayor.
Back to the point, though... Even though the Mayor of London doesn't have control over most tax revenue collected from drivers in London, this whole discussion is about what could be, so suggesting that the congestion and ULEZ charges are the only possible sources of revenue places an unnecessary limit on options.
With regards to VED collected in London. Only about £0.5 billion is collected annually from London. Fuel duty does seem to be broken down by region, so it’s hard know how much is collected in London. But across the UK £24 billion of fuel duty is collected annually, so it could be possible to fund a significant chunk of London’s transit by increasing fuel duty by 30-40% across the entire UK. But such an increase would likely also cause riots or similar. Additionally if you were to increase fuel duty like this, you would presumably need to provide free transit across the entire of the UK, which would require a significantly higher fuel duty increase. Of course that tax increase plus free transit, would result in a huge modal shift away from cars, and thus drive down the collected revenue.
In all, there doesn’t seem to be a viable way to provide free transit to all. At least not without significant tax increases across the board, and maybe that’s a viable approach. But there certainly isn’t an easy and obvious no-brainer way to get rid of the “bureaucracy” and use the savings for free transit, as was originally suggested.