I do this all the time. There is no big deal replacing the Ni-cds with Ni-mH of a higher capacity. The last pack looks like yours:
https://www.batteryspace.com/CustomNiMH3.6v2200mAhAAw/2APolyswitchand622AwgOpenWires.aspx
3AA and 2200 mAh with wire leads.
Yours may be AAA.
After doing it for many years (1975 for Ni-Cds) I was using tabbed batteries. Don't. get the ones with wire leads.
I have the part numbers for that connector and pins somewhere for your connector. I'm in bed now. You can "sometimes'` cut and splice with heat shrink. in an emergency. You can sometimes renew the pin housings.
I have a vtech phone that uses your connector and a vtech baby monitor that uses another, so I can find the part numbers/
The problem is the crimper is expensive. You'll pay about $70.00 USD for a cheap nice crimper.
I usually label the pack and it looks like I used the wrong company name. Name date, pack #. I always look for a battery with a warranty.
Used correctly, I've gotten 7-8 years out of a Ni-mH battery and they die at the same time.
A roll of Kapton tape comes in handy especially for those doors that like to fall off.
FWIW: They do cell the shrinkable sleeves.