(UK based question by the way)
If you have received a job offer, and now have the written contract, it's now time to hand in that notice to get the ball rolling for your resignation.
However you are on annual leave for a few days (3 physical working days). Your contract specifically states notice should be written and not verbal. Notice, in this instance, is specified in the contract simply as "one month".
Waiting out the extra few days means your start date inevitably moves forward.
If the new employer is keen on getting you started ASAP, what would be the best way to handle this?