What's your level? Depending on your skill, you might need different clubs.
1 Tabletennis Fightclub
Their members session are really good, you can find good player there. (normally Sunday afternoon) But you need a member's invitation at least.
Their open session where everyone can join, is amateur level. It is more of general practicing FH & BH drive/top spin, push, serve etc. Sometimes there are some entry level players just started to learn table tennis. So if you are good level player, you might feel bored.
I am division 2 player in London central league, and I can win some of their coaches in their open session.
But most of their members/coaches (more than half) are division 1 players. So their members' session are really good.
2 fusionttc / fusionttc.co.uk
If you are good skill player (good amateur or semi pro), this club is a must go. You can find session time on their website. It is almost open everyday. Around 12-16 tables there I can't remember exactly. Players' level are generally good. Lot of div 1-2 players. And it is open to anyone, just show up during session time.
If you are more of normal or low-level amateur player, you may find it a bit uncomfortable, as people tend to play against similar level player. (then the above open session will suit you best)
3 Moberly. http://moberlyttc.com
This is a really good club. Open mainly Thus, Fri,and Sat. The Fri and Sat session are really good (Charing Cross Sports Club). You can find various level of players there and really good.
There are a few div 1 level players, as well as a mix of div 2-4 or entry level players. Generally speaking, you can find a suitable opponents whatever your levels are. It is open to anyone, just show up. People there are friendly, and you can easily find an opponent. (unless you are semi-pro or pro player)
4 Morpeth TTC http://www.morpethttc.co.uk/
My friend told me that Morpeth is good club with good skill players. But I haven't been there, too far from where I live. If the time and location suit you, you can have a try.
5 Highbury http://www.highburyttclub.co.uk/
Again, this is another high level club in central london. Again, I haven't been their. It all depends on your time and location.
6 Barnet table tennis center (https://bttc.co.uk/)
as vvk1 mentioned, been there a few times. On weekdays you need to find someone to come with you as there aren't many players. Weekend sessions are fine. Sometimes there are pro level players but they train with each other. Other than those you can find a mix of different level players. (i will say from div 2-5 level)
And you need a car to get there.
Hope that helps.