jefmcg
Guru
On that strength, here's some stuff that might be useful
Decathlon is a french chain of giant sports shops that sell quality bikes that are really quite cheap. In fact so cheap that if you see them second hand they are possibly a bit over priced relative to the price they are new. There bikes are usually branded B'Twin.
Halfords are fairly common, you'll find them on most busy high streets ("shopping precincts") and retail parks. Oddly a combined car and bike shop. They sell some really crappy bikes (eg Apollo) and some quite high end ones (eg Boardman) as well as something in between as in the Carreras mentioned above. You wouldn't go far wrong with a Carrera Subway, for example. The shops themselves you need to be cautious of - some of the mechanics are really very good, but people have had awful experiences with them too.
Probably the biggest specialist chains in London are Evans followed by Cycle Surgery. They are OK, but hopefully you can find a local bike shop (LBS) near you that you like and trust better. The UK high street is dominated by chains, but independent bike shops are fortunately not that rare.
You can bring your bike on a lot of train and underground services outside peak times https://tfl.gov.uk/modes/cycling/cycling-in-london/bikes-on-public-transport which may be useful for getting it home. Helmets are not mandatory here.
Edit; or buy something classic from @biggs682
