CX is not obliged to give you an alternative flight at all. They may be kind to you and do so, but they have no obligation to do so. If they are hard-nosed about it, they can treat you as a no-show, invalidate your ticket and expect you to buy a fresh ticket, unless you are holding a fully flexible ticket that allows you to no-show but still use the ticket later. BA will not be responsible for your connecting flight either.

LCY is relatively prone to closure due to fog etc.

In other words, I think you should have a really good travel insurance that covers missed connection, or you should change your flight to the day before if the maximum connection time you can get on the day is 2 hours 10 min.