Hard to know what to say edboy! If a 12v battery charges up and then 'settles down' to 5 or 6 volts, it has to be very dead I'd have thought. A 12v battery is pretty flat at 10 or 11 volts usually!
I'm not sure whether that would be due to death by overcharging, but it's one possibility I guess. Were bulbs blowing, and was the ammeter saying anything useful?
Good quick check is to stuff a meter across the battery terminals - another one! - with the engine running and see what the voltage is at a fast tickover and above. You'd expect to see about 14v, staying steady-ish with increasing revs. If the voltmeter acts like a rev counter and the thing starts to show shedloads of volts then there is no regulating going on, so switch off to avoid damaging the battery or any bulbs you might be tempted to switch on.
I don't know about JGs overcharging, or its being well-known. I ran one for 20 years on one beast and it worked OK and still would if I hadn't put a DVR2 on instead.
If there is overcharging going on, first thing I'd do is check the dynamo to be sure that the D and F tags aren't touching behind the end cover, as that would bypass the regulator completely.
Having done that, I'd check the voltage straight off the A wire from the regulator to earth with the engine running to see what's coming through. Steady-ish 14v-ish would be right (and should be what you'd see at the battery too if there aren't any heavy loads, like the headlight, on).
If you suspect undercharging or no charge . . . If the bike is new to you, or the regulator is, I'd check the JG has been wired up correctly. JG regulators need the field coil wired from D to F, not from F to earth, being designed originally for what some people call 'series' wound dynamos like Millers rather than for 'shunt' wound ones like Lucas. I think JGs were first used by Vincent people back in the day on their Miller systems, so they were built 'that way round'.
The 'standard' Lucas dynamo connections are one brush and one field wire to earth, other brush to D terminal and other field wire to F terminal. With a JG it has to be field wires to D and F, one brush to earth, other brush to D. The basic difference is between regulating on the live side or the earth side of the field. In both cases, which wires/which brushes to which connection points will determine the rotation, so it matters. (The field coil wires are the two that come through from inside the dynamo, through holes in the brushplate.)
Don't know whether that helps or just confuses, but at least you have a reply apart from from yourself!