Just an additional datapoint here - in the past, I've had no reminder email to pay the balance on a BA holiday, and it was only when I logged in to check that I realised it was a couple of weeks overdue. Nothing had happened in that time, I paid the balance and took the trip as planned.

How flexible are your travel dates? I've just done a quick check and there are quite a few options around 22 April that are still available at BA's lowest prices, so you could gamble in the hope that they're still around in a few weeks for rebooking if you do decide to go. At least your dates are in the school hols!