the reg is almost certainly built into the base...
if it's a one hour charger it's going to be constant current with deltaTemp/time charge detection. NiCad show a large reduction in internal resistance when charging is completed, which manifests as a reduction in voltage required to maintain a constant current. Detecting that requires a "smart" charger, and a one hour charge (typically 1C) requires charge detection to be safe. The original charger was likely such a charger and, as you note, was being hindered by the replacement power supply.
The new charger, as evidenced by the small power supply, is a trickle charger from the start. These chargers simply apply a small current ad-infinitum - there is no charge detection. Typically the maximum charge rate for this is 0.1C for nicad. At that rate the heat from overcharging is dissipated by the battery fast enough that no damage is done. It should be fine as long as it states it's for an 18V battery and that the capacity of the relevant batteries is similar.
My biggest worry would be that the new charger could easily be for a new battery which has 2 or even 5 times the capacity, and as such is designed to produce much more current than the old battery can handle for a trickle charge. Since the power supply is only rated for 21V/200mAh it's relatively safe to assume it's OK.
That said, the OP should really find the capacity of the battery before giving it hell.