Oof, these people are master manipulators and their last message to you has really wound me up.
I'd lay good money on the fact that the leak has been there for ages (esp now you've told us their garden has been a state for two years) and they decided they'd get it fixed when the extension scaffolding goes up so they don't have to pay twice. So, whether it's crucial or not is a matter of opinion/further investigation, but not your problem.
Pleading poverty, oh dear god, please. How much is a 2 storey extension these days, 120k? If you can't afford another 1-2k to get cantilivered scaffolding on a two storey build, that's your fault for being financially irresponsible. Most home owners aren't naive or stupid enough not to build a 20% contingency fund into such big works. NOT YOUR PROBLEM.
Setting a deadline of Wednesday because they have building materials coming?Well, that was incredibly stupid of them, assuming you could be bullied into saying yes and then spending money. How every entitled of them.
ALL of this should have been flagged up at architect/initial builder discussion stage, so their lack of prepardness is really not your problem.
But mostly, I came to say this: I've seen manipulators like this before, don't even enter into a discussion about 'no because of my child or' 'no because I want to sell my house' - they WILL find a way to twist it so that you feel bad/they're trying to talk you round anyway.
Keep it simple with no reasons, e.g. 'sorry, I've considered it and my answer is no.'
If they come back arguing the toss, 'our poor roof' 'but we promised to make them behave':
'I didn't come to the decision lightly and I will not be changing my mind. I won't be discussing this further with you.'
No sodding way would they have the cheek to behave in this manner towards a man, how dare they.
Now, some people have rightly pointed out that you don't want to have to declare a neighbour dispute if you're selling, and they're very right to do so, but if I was trying to buy your house and you said 'they wanted to put scaffolding in my garden when I was in the middle of a separation in the height of summer', I'd go 'umm, neighbours are selfish dicks, be wary of them taking advantage, poor seller' and not bat an eyelid further. It wouldn't put me off buying your house. As neighbour disputes go, this one's really, really easy to explain.