For Farringdon your nearest place will be Ironmonger Row Baths. You can pre book or just turn up, no need to be a member. Gym with spin classes as well.
https://www.better.org.uk/leisure-centre/london/islington/ironmonger-row-baths
For long park run laps I’d suggest Regents Park. You can run the outer circle, which will likely have lots of central London cyclists whizzing around doing laps, or peel off inside.
River runs are fine. London is a very safe city. Head out south west along the river along the south bank as it’s more scenic than going east. They’ll be loads of other runners. I’d always suggest ditching any music though when running in unfamiliar areas.
Your nearest large Tri club is Trilondon. Always worth giving them an email, I’m sure they could accommodate you in some sessions.
Enjoy the trip. London is finally starting to feel like it’s old self post pandemic.