If you are talking about the London Cycling Club, their website looks like it would be a good club to join. The advantages of joining a club could include: learing from others experience, looking forward to club rides, and the encouragement from club members.

I will say contact a member to find out more information. They might be more receptive than you anticipate.

My compliments for taking the first steps. My first suggestion is to set realistic goals for yourself. No more than a 10% increase in distance each week on your bike, and remember to take some rest days for recovery.

Keep cycling and good luck!