He should ask a superior what kind of notice period he has to give by virtue of common courtesy but if there is no contract there is no obligation.
The employers could say "according to your contract" at this point "What contract?" should be asked. That is a double edged sword however. The employers could notice there is an oversight and give him a copy of the contract to sign (make sure it is dated the day he signs otherwise it could create retroactive issues). It could also create an air of hostility and if he asks about notice periods he could be sacked for any reason. Without a contract there will be no breach of contract but anything he could be dismissed for can not be constructive or libelous.
Any more issues contact the Citizens Advice Bureau who will explain the legalities of the situation a lot better. My friend is a volunteer at the CAB, employment solicitor and ACAS representative. Most solicitors have them and have a free half hour consultation to see if you have a case.