Should I leave NiCad rechargeable batteries charging when not in use for long periods

I would not charge a battery from other batteries; that is insane because it creates conversion losses for no good reason.

However, when one battery is charging, it is fine to tap that energy to charge another battery, using any of a variety of methods. Like any charging, this should be a proper charge controller, not just a diode array.

You don't want to be in a situation of leaving a battery partially discharged for any longer than necessary, because that ages a lead-acid battery.

Lead-acid batteries are a perishable asset

A lot of people, especially who live out in the country and have barn space, don't throw stuff out. They keep it around forever, on the logic that it "saves having to buy it again". Do not do that with lead-acid batteries. A lead-acid battery dumped in the back of the barn for 5 years will be dead as a stone when you go to use it.

Your only options for "surplus" batteries which are still working is a) sell it promptly to someone who can put it to work, or b) put it to work yourself.

So either get rid of it right now... or modify your system to make use of its capacity on a daily basis. Every battery is an addition, because discharge wears lead-acid batteries, and the rate of wear is an exponential function of discharge depth. Draining 2 batteries to 55% full is much worse for total-cost-of-ownership than draining 3 batteries to 70% full.

"Spare for when one runs flat" is also a fail

Similarly, the idea of running one battery pack dead, then switching to the next pack, is a bankrupt idea as far as total cost of ownership. Running a lead-acid pack to failure is absolutely brutal on service life of the battery. A car starting battery is destroyed by 5-10 dips to dead. A battery specifically for solar is destroyed by some dozens of such dips, and worn/damaged/capacity reduced by every single dip.

Again, for best use of the batteries, you want to pull them down as evenly as is practicable. 80-100% won't matter, so pull A to 80%, then B to 80%, then C to 80%, then A to 70%, B to 70%, C to 70%, etc.

Even better, have a smart battery merger/controller that blends all packs together, drawing them down equally; however this requires some intelligence - diodes alone are not enough. (Also diodes take a bare minimum 0.2V drop and most diodes are more like 0.8% -- that's 2-6% of total battery capacity being wasted in heating up diodes.) A manual string of switches (B on, A off) would be more efficient.

Physical topology is also a consideration

If your house is spread out, and 60% of your load is in one place, 25% in another place and 15% in another place... it may be better to distribute your packs around the house so that the batteries are near the loads, rather than "wheeling" (hauling) 12V over long distances. Voltage drop bites 12V very hard: with only 1/10 the voltage of 120V, it has 10x the voltage drop, which means 100x the power drop. You can end up losing a significant fraction of battery capacity to wiring losses, unless you're up on the ought-sizes of wire.

Get a hydrometer and learn how to use it for detailed analysis.

For less detailed analysis, if a lead acid bank is below 2.1 volts per cell (in use) the smart money is on cutting it off if you care about long life. So if your 12 or 24V bank actually reads 12 or 24 volts, you are probably dipping below a healthy discharge level.

Better quality battery operated equipment will disconnect itself, rather than depend on you checking the voltage and taking it off-line. But we have no idea what your inverter is since you have not told us, and if it's a lower-end model it may not have a very good behavior for when it "alarms." (Even with the model information it may be hard to know unless it happens to be a well-known quality model with good documentation. Cheap junk tends not to document the things that reveal it to be cheap junk.)

Much better quality battery operated equipment will record the number of amp*hours put in and taken out (and know how big the battery bank is) rather than using the voltage as a rough yardstick.

This website provides some more food for thought, but practical and perfect don't tend to get along well in real life battery bank use scenarios. i.e. you are unlikely to disconect it for 4 hours before checking the voltage, at least in normal use. Occasionally for better numbers when doing maintenance, perhaps.

...And voltage on the whole bank does not give you the detailed info that checking each cell will - if you don't have 12 individual 2 V batteries making up your bank, the only way to get that into is with a hydrometer.