Well the first thing to check is your policy.
Your insurance may still be valid without tax. But different providers have different rules.
Assuming your car is drivable, you'll need to get your car taxed pronto anyway, or declare it off road.
There is no official grace period any more. Do it online now.
Late payment fine is £80. But there's a good chance you wont get penalty at all. Worst case any penalty will be minimised.
I've taxed cars late before - but within a month of due date - and had no follow up. The DVLA, I'm sure, are more worried about habitual offenders etc.