For the sake of accuracy, it wasn't a made up letter, just misunderstood (or misrepresented if you prefer). The obligation (strictly just a condition of funding) was for expanding the ULEZ to the N/S Circular. However, there is an overriding legal obligation to do something about air pollution in the whole of the Mayor's area.
"A local authority must, for the purpose of securing that air quality standards and objectives are achieved in an air quality management area designated by that authority, prepare an action plan in relation to that area." (Section 83A(2) of the 1995 Environment Act - a 2021 amendment to the Act - so a Tory amendment to Tory legislation). This is,
I think, the legal obligation on the Mayor to do something about pollution, of which older vehicles are a significant cause.
The government had been carefully neutral about further expansion (or, if you prefer, careful neither to support or oppose it, just to say it's up to the Mayor).
See the later Shapps letter I referred to, and this parliamentary answer: