Whilst they can make any stipulation in a contract that they like, whether it’s enforceable or not is a different issue. Sometimes, a contract will have a provisioning where the employee cannot work for a competing employer for a period of time. It depends on the time period but just because that maybe in a contract doesn’t mean that it’s enforceable.
This is no different. Just because it says 20 minutes does not necessarily mean that the employer could enforce it. It would be different if for example it was the fire service and you are required to be there within a particular period of time in the case of emergency from what you have said, this seems to be no reason and they are not prepared to give one. Hence, if you decide to ignore this and simply move anyway, it’s most unlikely that they could get rid of you breach of contract. The breach is not fundamental, it’s minor, and it doesn’t strike to the heart of the contract. It just seems to be some arbitrary provision from what you have said.
Unfortunately, if they decide to get rid of you, you cannot pursue a claim for unfair dismissal until you have worked there for 2 years (it was 12 months before 6 April 2012). There are certain reasons when you can pursue a claim for unfair dismissal within that period such as discrimination, health and safety, whistleblowing et cetera but this alleged breach of contract is not one of them.
If you think they are doing this to try and get rid of you, and making things so uncomfortable that you feel you have to leave, then you have a claim for constructive dismissal but unfortunately you cannot bring that until you have been therefore 2 years at least.
If they get rid of you for this reason you will not be able to bring a claim for unfair dismissal you may have a claim for breach of contract.
If he does dismiss you, you would have no option but to sue him for breach of contract. Some solicitors would deal with this on a no win no fee basis and it would be worthwhile certainly speaking to a firm near to you to see whether they would be interested in taking this on.