It is not uncommon to see cars that have been registered in the UK have a small section on the left-hand side of the number plate that will indicate that the vehicle is registered in the UK. For example: Will this identifier be removed from vehicles that already have it on their number plates, or will cars still have this identifier after Brexit?
My car came with the "EU flag and GB" on the number plates many years ago. When recently, after the Brexit referendum, I needed one replacement plate (the other wasn't damaged) I specifically asked for one with the "EU flag and GB". The garage struggled to locate the correct blank template, but eventually found one for me - apparently they were no longer able to get blank plates with the "EU flag and GB" so had to hunt around in the stores for one.
I have noticed in recent years that fewer and fewer GB registered cars have the "EU flag and GB" on their number plates.
The point of this... I really doubt that my car's number plates are suddenly going to become invalidated with Brexit, just because they have the EU flag. Nor do I think the police/DVLA are going to be enforcing new number plates on people. However, if I was to need another replacement plate, I think it might become impossible to get the "EU flag and GB".
When traveling in the UK I do see more cars without the EU/GB tag than with. (In other EU countries there is a 95% + number of EU tagged cars it seems, in the UK it seems about 5%.) So I think there will not be that many people who will have to get a new set of license plates.
But as cars with plates from before their countries joined the EU still use them years later, I do not think it will be a thing which will have to be done with high speed.
As with all Brexit related things, we can only guess for now.
I assume that those people who have the special EU plates and want to get plates without the tag will have to pay for their own.