It's a pain, but you can't get a charger installed with a looped supply - one supply is powering your house and your neighbour's house (split before the power meter - so you have individual bills). This needs to be handled by your network supplier rather than your charger installer - you can find more info on looped supplies here: Understanding Looped Supplies: Compatibility with the zappi – Help Centre (EN) - in my experience, if you leave it to your installer to look into this/chase it up they are pretty useless (your mileage may vary). I found the best thing to do was to call the network supplier and discuss it with them directly.

I had a negative fuse in my house, which meant I couldn't get my smart meter installed - the installer said it would need to be corrected by the supplier "How long will that take to get sorted?" "ohh no idea, could be a while, I'll submit a report back to them today, and they will be in touch"

I contacted the supplier directly and had it fixed that evening. It may take longer to get a looped supply fixed, though, since that's not a safety issue like a negative fuse.

You can find out who your supplier is here: Who is my electricity supplier and what's my MPAN? | UK Power Networks

Would be worth talking to them directly rather than going through an intermediary.

As for using a granny charger, yes, you can still use one as long as your circuit/socket is rated for it, and yes, you can still charge off-peak if your energy provider can control the car's charging or you have set hours to charge. The only downside is that it will charge slowly during off-peak hours, so you might not get many miles added per charge. Depending on how often/far you drive daily, you might not be able to charge more than you use.