That's complicated:

* Buses (outside London): yes. Private and (mostly) deregulated.

* Buses (inside London): kinda. Private, but heavily regulated (fares, routes, everything)

* Trains (outside London): both. The infrastructure is now public, after the company it was sold to became insolvent and got essentially renationalised by stealth. The companies running the trains are mostly heavily regulated private (some fares, routes, and other things are fixed by their franchise contracts, some fares are flexible). A few companies are 'open access' (private without franchise) and only their routes/timetables are fixed, they can do their own thing with fares and rolling stock. A few places (such as long-distance trains on the East Coast mainline) are run by a subsidiary of the state directly.

* Trains (inside London, including the Tube): Infrastructure also public (some run by Network Rail, the national organisation, some by Transport for London). Some services (e.g. the Tube) run directly by TfL, some franchised as nationally.

* [Edit: Eurostar runs (lightly regulated) services to France and Belgium via the Channel Tunnel and sets its own fares]

The main difference from other countries (except for the insane web of contracts and inter-organisational dependencies) is that the policy of the UK Government is to put the cost of public transport as much as possible on its users rather than subsidising from general taxation. We've had above-inflation rises in the regulated fares nearly every year since privatisation in 1994, for example.

The public transport politicians use (London Underground, London buses) is publically owned.