The Mayor has today published the TfL business case for taking over more suburban rail services. This is a pretty substantive document, 99 pages, with costs, benefits, extra train numbers and future service patterns and frequencies all set out. The investment needed to support the improved services is also set out. This has gone to the DfT so it looks like something is going on - there is also a pretty decent list of supporters and supporting letters.
https://tfl.gov.uk/cdn/static/cms/documents/rail-devolution-business-case-narrative.pdf
I know the forum rule is to cut and paste the document but 99 pages is rather too much so I hope I can be excused from breaking the rule on this occasion. Here is the summary.
Interesting how it basically ignored the South West Lines. Zero speak of the SW stations, services, rolling stock, anything really.