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If you can convince upstairs to fix it without going through your insurer there's nothing wrong with that, it's just not guaranteed. You'd have to rely on your powers of persuasion, intimidation, or your upstairs neighbours ignorance of the law and willingness to pay AND to do it to a proper standard. If those things are a possibility it's worth a bash.
However I've been in this situation multiple times and the only thing I've found that works consistently is going via your insurer so they can contact your neighbour and take care of the problem. Either your neighbour has insurance in which case the companies will know what to do, or your neighbour doesn't in which case your insurance company can chase them through the courts for remedy. Either way the problem isn't yours anymore, and that's ultimately what you pay for with insurance.
Fwiw when I did this, I didn't have to pay an increased premium. Suspect this might happen to people with no claims discounts, who then make a claim, but I think that's about it.
That's how it works.
You go through your insurer. They chase the insurance company of the people upstairs or chase them directly if they are uninsured. Either way your insurance means you should be guaranteed to get a payout for it.
Of course your insurance premium will go up because you're more of a risk, even though it was someone else's fault. Plus you have to notify any other insurance company of the claim, and possible buyers if you sell your property in the next 5 (I think) years.
It's all better than having to pay for it yourself though.