There is a grace period, especially since the UK hasn't even started the formal process of leaving, yet. Parliament will also need to decide on this.

There is only one scenario without a grace period: some more radical Brexit supporters advocate breaking EU laws and regulations immediately (right to work and live and so on), just because the European courts take their time and by the time they'd hear the case the UK would already have left, so no consequences.

In that case some European politicians have replied there will be an immediate cutoff from the Common Market.

Both threats are highly unlikely to be played out, IMO.

It will officially take two years to leave once the process has started, which it has not been yet (that's what the Prime Minister declares). Whether or not you subsequently have to pay import taxes after Britain has left is also something that will have to be negotiated.

Nothing has actually changed yet.