You can be fined up to £1,000 for driving a vehicle without a valid MOT. You can get an MOT up to a month (minus a day) before it runs out and keep the same renewal date. If your MOT runs out on 15 May, the earliest you can get an MOT to keep the same renewal date for next year is 16 April.
You can't extend your MOT date, only move it forward. You don't want to drive around without a valid MOT after all. You can book your MOT up to one month before your vehicle's existing certificate runs out minus one day. So this is the timeframe you can change your date.