When does TA become binding on LL's part?

Hi to all.

I know that T can pull out till the last day. Only when they have moved in, the TA comes into force and makes them responsible for rent for the duration of the TA.
However, as I understand it, it’s not the same for the LL, who, once deposit received and TA signed (we do it several weeks in advance), is bound by it.
We had a potential T who presented very well during the viewing but then showed a different side of himself and said he couldn’t pay first month rent on the day because “it didn’t work for him”. We have already signed TA, Luckily, we manage to agree to go our separate ways.
In another case, T demanded to move the start day to a week earlier at two days’ notice and got really rude telling us “to make an effort for the money”.
Can somebody confirm that T and LL don’t have equal rights in this situation (TA signed, security deposit paid), and T can pull out till the last day, but LL can’t.
In both situations we really wanted to let them go, even if we were losing the money.

Youve conflated two things. The contract and the tenancy. The contract, (tenancy agreement) is binding on both parties once its in force, which is usually when its signed, but not always. Once a tenants signs a TA they are bound to pay the rent for the duration of the tenancy. However, at this point they dont have a tenancy. That happens when they take possession, (move in). If they have a contract but no tenancy, then the landlord is required to mitigate his losses. Usually that means finding someone else and releasing the existing contract holder from their obligations. Once there is a tenancy, that requirement disappears and the landlord can legally demand the tenant perform the contract in full.