If the battery has been left for a while, maybe the internal plates have started to 'go' (lead sulphate hardening on the plates, I think)
To test the battery, the best way is to ask someone with a high-discharge tester to put a heavy load on the battery and note the voltage (which should not dip below 9.5 - 10V, but that depends much on the tester itself (some are heavier loads than others for use with larger batteries)) and notice if any of the cells are rapidly bubbling up (which means dead cells!) This test is best if the battery gets an 80% charge
The other way I have seen to test the battery is to disconnect the coil wire (preferably the wire on the -ve (assuming -ve earth!)/CB side of the coil going to the dizzy (I think it's white with a black tracer, and possibly has a male spade on the wire (our series 2 2000Auto did))
Put a voltmeter across the battery, then have an assistant crank the engine over for about 10 seconds. The voltage during cranking should not dip below 9.5V (ideally not under 10V, but batteries that are 'nearing the end but are still serviceable for the time being' seem to go dip below 10V slightly.) It's likely any dead cells will rapidly bubble up, but is not always the case when using this 'voltage under cranking' test.
To test the charger:-
This is not a robust test as it only really picks up if the charger is pumping out a voltage, and is not really recommended:
You could put a voltmeter on the terminals while it is not connected to a battery (be careful the crocodile clips do not touch each other, of course!), but this does not always work as some chargers 'sniff' there is no current flow and so cut off the supply to the battery leads. (Quite sophisticated transistor switching!)
If you do go down that path, and there is a voltage on the clips when not under load (i.e. when not connected to a battery), the voltage should be in the region of 13.6-14.5 V (for a trickle charger), higher (15-15.5V) for a 'quick' charger. Different chargers have different off-load voltages, and are not easy to spot.
The better method would be:-
Take a voltage reading of the battery (which should not be less than about 12.0V). Record/remember that voltage. Put the battery charger clips on, but do not switch on the charger yet!. The voltage should be the almost the same as before. If it's significantly (> 0.5V) lower, it suggests the charger diodes are blown and are 'leaking'.
Now, turn on the charger, and notice the voltage reading. If the voltage reading is the same as the reading when the battery was not connected to the charger, it suggests that the charger is not pumping out any juice. If the reading shoots up to 13.6V - 14.5 V, the battery is not accepting any charge. For a trickle charger the reading I would think should be in the region of 12.3 - 13.0V for a flat (0%) - 25% charged battery, 13.0 - 14.0V for a 25 - 75%, 14.0 - 14.5V for a 75+%
It's no way an absolutely accurate guide, I hasten to add. Very many factors influence these tests!
[A general rule of thumb: Battery offline voltage = 12 + (0.6 x amount of charge)
Battery 25% charged, offline voltage = 12 + (0.6 x 0.25) = 12 + 0.15 = 12.15V
Battery 50% charged, offline voltage = 12 + (0.6 x 0.5) = 12 + 0.30 = 12.3V
Battery 75% charged, offline voltage = 12 + (0.6 x 0.75) = 12 + 0.45 = 12.45V
Battery 100% charged, offline voltage = 12 + (0.6 x 1.00) = 12 + 0.60 = 12.6V
Again, this is not totally accurate, merely a guide, I hasten to add!]