He is entitled by law to 4.8 weeks paid holiday per year (24 days), including statutory, bank and public holidays.
He is also entitled by law to a written statement of employment particulars. The contract was made as soon as he accepted the job offer, so he does have a contract of employment.
However, actually getting his full entitlements is another thing altogether. It would be very easy for his boss to dismiss him - he would only have to say that there is not enough work to justify employing him.
He could take it to a tribunal where he would win but it is doubtful he would have a job at the end of it all.