No, you do not have to change your number plate. But don't let the small matter of the facts get in the way of another Brexit imagined issue. You simply put a UK sticker on your car. If you did want to change your number plate, simply get one with a cross of St George on it, and that should nicely future proof you as well as being legal.
There is no requirement to remove the EU flag from your number plate. In fact I have a car, with number plates, and have no flag on them at all.
That is most wise, especially as you don't have to.
What I don't understand is that you are all going on about this huge issue, but no-one is mentioning that the rules for driving in France, but no other country, is that you need to carry a breathalyser and a breathalyser is about £5 instead of £1.50. So you have to do something different in one country to all the others and you haven't batted an eyelid, you have to spend a couple of pounds on a sticker valid everywhere and it is some major bloody trauma!