Firstly, there is not just one pre-action protocol. There is the Practice Direection on Pre-Action Conduct and Protocols and then there are specific protocols which apply to particular types of claims. Per paragraphs 2 and 6 of the Practice Direction, if a specific protocol exists you must follow that, otherwise you follow the general one.
For the sake of this question I'll assume you are following the general protocol. In that case there is no fixed period of time. The protocol simply states at paragraph 6(b) that the defendant must respond within a "reasonable time" which is somewhere between 14 days for a simple case and 3 months for a complex one. It's up to you to decide how complex you think your demand is and what is a reasonable amount of time to allow the defendant for a response. Personally for most non-urgent cases, I like to err on the side of caution and give 1 month with a further 14 day reminder in the event of no response, but this is not strictly necessary if the claim isn't complex.
To re-iterate, it's very important that you check whether a specific protocol applies. E.g. for debt claims pursued by businesses against individuals, you must allow 30 days for a response.