Nissan Leafs show their battery health on the dash - 12 bars when new, down to 3 or 4 if someone has really ragged them.
Yep, what @Ionacat said, you can get detailed health checks - should show even wear across all the cells. A bad cell shows up exactly like you said - sudden drop in estimated mileage, particularly when you hit the power for an extended period.
If you want to test a Leaf, take it to a fast charger, charge to 100 percent and then find a road where you can max out the power for a bit (steep hills ideal). Should bring out any bad behaviour from the battery.
I would also try to buy from an EV specialist. Ask what warranty you can expect on battery failures.
For £8k you could probably find a 40kWh Leaf - assuming a bit of wear, maybe 110 miles in summer and 70 in winter? Or you'll find loads of 24kWh models (like mine!) that are more like 70 and 40 miles.
ISTR the short-lived 30kWh models were a bit cursed, I'd avoid those. And 60kWh is probably outside of that budget.
You might not need a charging point depending on your daily mileage, just get an external 13A plug socket fitted, maybe £100. (If you know you'll need faster charging, I'd budget £1200 for a simple install).
Also check your electricity supplier and get onto a cheap overnight tariff (e.g. Octopus Go).
The Reddit Leaf forum is pretty friendly for more info.
This doesn't apply to the Leaf but I wouldn't take on a battery lease. But I hear buying them out is straightforward if you do find a bargain Renault Zoe.