Any organisation which is classed an association (see below, Govt guidance) is covered. IANAL, but I think that if you start any organisation or club it has to be open to everyone on the same terms unless the reason for setting it up is to provide opportunities for a group that share a protected characteristic. If the club is just
The Act defines an association as an organisation that:
• has 25 or more members, and
• has rules (not necessarily formal or written) regulating who can be a member and there is a genuine selection process for members.
Examples of associations include private clubs such as golf and other sports clubs, ex-forces clubs, alumni clubs, social clubs, working men’s clubs, gaming clubs and drinking clubs.
If the cycling events were just pay and turn up they might not be covered, but if they have genuine membership and racing categories split by gender then that might count as having rules? It depends how they are set up, but basically if they do fall under the Act and categorise by gender identity, then they are indirectly discriminating against women and also any males without the PC of GR and perhaps anyone who doesn't think they have a gender identity or is religious (PC of belief). Please don't take my word for it, because IANAL. But I think that is the general idea. There's also an official. human right of equal access to sports for girls and women.
I do hope this isn't garbled nonsense. I've got post-viral fatigue and my brain shut down a couple of hours ago!