I don't think London has private mini buses like that, just a huge amount of buses.
They are operated by companies for TFL (though they are all red, you can read the company name).
I'm not sure at what point that arrangement happened, somewhere between the outright privatisations of the 80s, or the stealth ones of the black years, which used PFI.
I think citymapper ran an experimental private bus line because they had identified a gap. But their article doesn't have a date, and I seem to remember this was years ago, so not sure where it's at.
https://citymapper.com/news/1800/introducing-the-citymapper-...
The existence of private minibuses is a clear indication that the transit authority has failed.
The experience of bus deregulation in Britain outside of London tended to go one of two ways. In most of the country it simply created a private monopoly while in a few cities like Manchester there were routes so popular (busiest bus route in Europe!) that multiple companies competed for custom and the city council had to employ stewards to ensure fair play.
This level of success was really an admission of failure. Anywhere else would have built a subway or at least a tram. (Metrolink is great it just doesn't even attempt to serve the busiest transit corridor in the city.)
London was of course treated differently. While the operators were privatised they remained regulated by government. Public transport in London is simply too important to the UK to be allowed to fail as it was in other parts of the country.
Anyway it's good to see central government allow Greater Manchester to regulate its buses again. After smashing it up local transport in 1986 the Tories finally relented in 2023 and allowed Greater Manchester control. Quite what Labour were doing for local transport outside of London between 1997 and 2010 is open to question. But it certainly resulted in renaissance of London public transport.