You do have a contract - you just don't have it in writing.
However, your employer should give you a written contract within 2 months of staring work, setting out the terms and conditions. Ask him again for a written contract.
As you don't have a written contract, if there is a staff handbook your terms and conditions will be as stated in that.
If your company doesn't have a handbook, then the industry standards and the law will apply.
You will still be entitled to 28 days paid holiday, redundancy pay, statutory sick pay, a right to claim unfair dismissal exactly as if you did have a written contract.
Submit a request in writing for your written terms.