It is extremely unlikely BA will allow you to change by a day for free especially now that it appears you have already accepted the change. Never just accept a change of flight times that affects other, connecting, flights. Look into alternative flights before you accept the change,

The MCT is not just about the time check-in closes (that is 45 minutes for short hail at LGW) or even boarding closes for the next flight. It is to allow you time to get from your arriving flight gate to the departure gate whilst completing any additional formalities e.g. passport and / or security check (plus for any checked luggage to get there too) plus some spare time to allow for a late arrival.