How related are NiCd and NiMH chargers?

Its possible but hard to answer without knowing the charger topology.

NiCd can be handle very high charge/discharge currents. Such charge currents could damage a NiMH battery.

Now it MAY be OK, not all NiCd chargers are of the high current/peak detect variety, sometimes called "Rapid Charge".

Some are simply timer based in which case they will very likely undercharge a NiMH battery.

Still other NiCd chargers are "trickle chargers" made to slow charge overnight, such a charger would work with NiMH but would likely take a very long time to charge since NiMH batteries generally have a much higher capacity.

Yet another charger type is the "smart charger" which monitors the battery as it charges and adjusts its charge rate, stopping or going into a trickle mode when the battery is determined to be full. The "full" trigger and charge profiles are different between NiCd and NiMH so I would assume this type would need to be modified (possibly with new firmware) to handle NiMH batteries.

EDIT: Ancillary question: are you sure switching battery chemistry is a good idea for this application? The mAh rating doesn't tell the entire story of battery life. If the device is high current then a NiCd battery may last much longer than a NiMH battery even though the NiMH battery likely has much higher mAh rating.

NiCd and NiMH batteries do have similar voltage characteristics. A NiCd charger might do OK to charge NiMH batteries, but I wouldn't assume that without knowning what strategy the charger uses.

NiCd are a bit more forgiving than NiMH in charging. They can generally take higher charging currents and are more tolerant of trickle charging, although even most NiCd don't want to be trickle charged long term.

The end of charge is trickier to decide with a NiMH. Most NiMH chargers are therefore "smarter". I've seen some NiCd chargers be as dumb as a transformer, full wave bridge, and resistor. That's not so great for NiCd, but even worse for NiMH.

All in all, I'd not use a charger supposedly for one type of battery on another.