This is the relevant part of the UK Conditions of Carriage:

3e) Your rights if you are prevented from travelling by events beyond your control

If:

you are a consumer.
you have been prevented from travelling by events beyond your control and
all or part of the fare for your ticket is non-refundable.

we will give you a credit for the non-refundable part of the fare. We will do this if you:

have a completely unused ticket
have told us promptly about the events beyond your control and
have given us evidence of these events.

The credit can be used for future travel on us by you or any person you choose. We may take a reasonable fee from the credit to cover our administration costs.

As SRG says, you can't travel while the child is still infectious. If you are due to travel within (I think) six days of the appearance of the rash therefore you will not be able to travel. BA are reasonable; I suspect that they will say that the child and one or two family members (eg Mum and Dad) can vary their ticket in accordance with the conditions of carriage. Anything more, and you may have to rely on travel insurance. But as SRG says, best to talk to BA.