It is perfectly possible to have a CIC with just one member, and I have designed and registered a number structured like this. If that sole member is itself a corporate body then the single-member CIC is a “wholly-owned subsidiary” as defined in s.1159(2) of the Companies Act 2006 and certain consequences flow from that (e.g. consolidation of accounts), so you may want to consider these consequences before proceeding.
The directors of the parent CIC will control the single vote in the subsidiary so they will appoint and remove directors as they see fit. If they choose to delegate that power somehow to the members of the parent CIC, e.g. by asking the members to elect directors of the subsidiary, that’s up to them, but it will be a delegated power and may be withdrawn.
If the subsidiary status of the second CIC doesn’t sit well with the Articles as registered, e.g. because they allow for multiple voting members, then it’s a good idea to have the Articles re-drafted to make explicit the fact that it will in future have a single corporate member. There’s no barrier to doing that.
These observations apply whether the CICs are limited by guarantee or by shares, or if there’s one of each.
Thanks Charlie. That's really helpful. Looks like I have some more research to do.