Okay. Technically when you signed the lease you agreed to its terms which will have included payment for rent and may also have covered a notice period (but this lease is very short). However, you should also have been given info to be enrolled in a Tenancy Deposit Scheme - which they have not done. This is a legal requirement for residential tenancies so they have not complied here.
Whilst technically they could enforce the payment covenant and insist you pay, with such a short lease I am not sure it would be worth it. They would have to bring a claim against you in the Small Claims Court.
On a practical level - since no money has changed hands, since there's no arrangement for a deposit, I think you are within your rights to say you don't want it any more, there has been a change in your circumstances and add what else you might want to say.
So, the quick answer is yes they could try to enforce payment if you have signed a lease with a payment clause in it and you have not given them notice (Not sure whether this is in the lease), but it's worth trying to pull out. I would make it clear that the transaction has not been completed due to non payment of any deposit and monies and not been advised on the Tenancy Deposit Scheme.
I hope this helps. If not, do please ask.