Your tenancy agreement will probably say that you need to allow access for viewings, etc.
I agree that you should try to get the new landlord to agree to a moving date 14 days into the future. Tell the EA that you cannot afford double rent, and your current landlord demands rent to the end of the lease.
I have just gone through exactly the same, and had to swallow a very bitter pill. As long as they have the law on their side in regards to this, they are unlikely to budge. Your only leverage is that a 21 has not been served, so legally you have the right to remind in the property (and pay rent) until 2 months after a proper formal notice has been served.
If the new landlord is unwilling, you can suggest to your current landlord (if you want to):
Dear Landlord, I dont mean to be difficult and I understand your situation in wanting to sell the house and that you require my rent to cover mortgage payments until the property has been sold.
I have now trouble securing my new property in this time frame, as I will be short for rent because the new landlord wants the lease to start mid month, which means I have two rents to pay.
It has come to my attention that I have not been served with a formal notice 21, which is required by law when giving a tenant notice. In order to solve the financial problem for both of us, may I suggest you serve a proper section 21 notice, which will give me 2 months to find a new property and adjust our moving date accordingly to tie in with the end of the lease as per the formal notice. This way I wont have to pay double rent, and you get rent paid until the end of the tenancy.
I know it is a real pain, this will mean it is another two months until we vacate the property, but neither of us will suffer any financial hardship this way.
Kind Regards,
Kindness