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I had a think about doing this previously with just a bog standard UPS like this
https://www.amazon.co.uk/APC-Back-UPS-BV1000I-MS-Uninterruptible-Universal/dp/B07Q9KPHLX?th=1But then I thought what would the point be? If I didn't have any power then none of the linked stuff would be working anyway. No point keeping Hue hub going for instance because the lights all need power, most of the stuff that use the internet wouldn't have power either so why bother keep internet going?
FTTP is powered by electricity at the exchange and in the local cabinet so any loss of local area electricity and that will drop out. 4g likely to go the same way if the power drops in your local area (happened to me recently)
The question is, what are you hoping to protect against with this battery? a home power loss (distribution board packing up) or loss of the local area supply.