An employer has no legal obligation to supply a written contract. However, they do have to supply 'written terms of particulars' within 2 months of your start date.
You have had a verbal contract from day 1, which is just as legally binding.
With regard to notice period, alligator is correct.
Technically, if you choose not to give notice, then they can sue you for breach of contract, but only for the difference it would cost them to replace you for your notice period