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No, even in the worst case scenario regular tourists would not be forced to leave the country. Non-EU nationals entering the UK normally get a 6 month leave to enter and there's no chance EU nationals who have already entered the UK will be treated worse than that. So if you want to be safe, make sure your trip doesn't last for more than 6 months since the date of your original entry.
NB: Nobody knows for sure what's going to happen on Brexit day. Treat the above answer as speculation and check the news on March 29th to get the full answer.