There is no compensation if the cancellation is communicated 14+ days beforehand, as is the case here. However you are entitled to all the other UK261 rights - e.g. you can choose to get rebooked at another date of your convenience. You do not have to accept a rebooking 2 days earlier.

If the issue is that BA no longer operate a flight on the date you booked, you would be entitled to be rebooked on any other airline(s) which offer a direct or connecting flight on your booked day of travel. BA will undoubtedly refuse to rebook you if it's outside their quite limited internal guidelines, but those guidelines have no bearing on your legal rights.

The CAA guidelines (see paragraphs 4.2 and 4.3) on interpreting "at the earliest opportunity" are clear - if an airline can't offer a same-day alternative on their own flights, they must offer to rebook you on any other airlines that offer a same-day flight.If BA are refusing to do so, you'll ultimately have to make your own booking on another airline and then claim back the cost after you travel. Unfortunately this probably won't be a quick or easy process and if this is the outward leg of a return trip, you might face issues with your return leg since they could say you were a 'no show'.

What's important now is to not accept a refund or any alternatives that you don't want to fly on. Try to get any refusal to rebook you the same day in writing - this will make it easier to claim back your expenses afterwards.