The agency can create a legally binding tenancy agreement without the landlord signing the tenancy, as long as they have signed terms and conditions between them and the landlord that allows them to do so. It's actually quite rare to have a landlord sign the tenancy if they have instructed an agent.
In my mind I am thinking that either your landlady is a bit slap dash and has instructed more than one agent to let the property (which is not unusual) and has received offers from both agents and has failed for whatever reason to tell one of the agencies not to go ahead with the deal.
The other more likely scenario is that your agent have had some sort of breakdown in communication between them and the landlord and have just assumed that once the offer has been initially agreed that they have the green light to just go ahead and move you in on that date, probably using the set of keys that were provided to them for viewings when they were instructed.
But even that really don't make too much sense as the agents would have been in contact with the landlord during the referencing process to let them know how things were progressing, they would have had to have written to the landlord confirming the terms of the deal, they would also have had to get bank details to make payments into or to enable them to set up standing orders if they were not managing the property.
Also, has the property got gas in it? Has the agency obtained a gas certificate from the landlord, or did they have to speak the landlord to get instruction to carry one out?
There are quite a few reasons for the agency to be in contact with the landlord during the referencing process that it baffles me as to how this situation would arise if the agents had done their job.
On the other hand, people are people and your landlord might be a nutter and has got completely confused and forgotten they have agreed the two deals?