This will depend on your ticket conditions. But generally, on a typical non-flexible ticket they are likely to charge you a change fee of £60 or £100 plus any fare difference for the change.

Best option is really to call them and ask them. If you do not like the cost and cannot claim on your travel insurance, you always have the choice to politely decline the change and stick with what you have and perhaps take some medicine to reduce your symptoms.

TBH, for what sounds like a cold, both BA and travel insurance are unlikely to be particularly helpful.