Hello, I am a solicitor with over 15 years experience. I will try to help you with this.
Have you and the agent/landlord signed the new tenancy agreement?
OK if you have signed and paid a security deposit you can reasonably treat the agreement as made. Did the agent sign on behalf of the landlord. I don't think it is necessary for there to be a signature from the landlord for the agreement to be binding.
There is a start date, you have signed and paid a deposit this was with the agents present I assume and unless you hear to the contrary you can move in on your start date.