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But their argument - they'd have to find rental accommodation for this person in the meantime
.....
they're an organisation which owns 400+ properties, including some they let out
Please do not allow yourself to be bullied by these people. They have 400 properties - their incoming tenant will be ok I suspect.
They want to buy your house and don't want to start from scratch with another. That stands to reason.
I'd say you need a week or so to try and see what arrangements can be made and then say oh sorry it's not possible but we will keep vigorously trying to either complete on our purchase or find a rental in the meantime. The longer it takes the more committed they are and the more likely you are to get their way.
I'm intrigued who they are tbh.
The solicitors tried and got a hard no which contained the word "unacceptable" and said that it was us who'd been pushing for the quick sale (definitely not the case).
The estate agent tried with the buyer and they got a two-week extension which I think we're supposed to be grateful for.
Their story is legitimate, they have someone they want to move in in August and there's work they want (need?) to do first. We knew all of that at the start (apart from when they wanted someone to move in, which we would have said was going to be tight). But their argument - they'd have to find rental accommodation for this person in the meantime - is the same as ours, but we're a young family of key workers trying to relocate, and they're an organisation which owns 400+ properties, including some they let out, so it should be easier for them to compromise than us.