As others have said, it it is possible to solder onto an untagged NiCd or N iMh battery. But it is also easy to get hold of good quality tagged cells f or not a lot of money. These save a lot of grief and the possibility of dam aging the cell - it may work after your soldering but eg. the capacity may be impaired.
I did exactly what you are talking about, replacing the battery on a Braun rechargable shaver. My model is the 5 476. I got a battery from the 'Small Battery Company', who are quite good for small orders and have a big range:
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I think I probably fitted an NiMH AA tagged cell. There is a knack to takin g them apart, but it sounds like you know this. I managed to get hold of a service manual which helped.
My shaver, like yours I suspect, is not a fast charge one. So far (~6 month s later) it seems fine, and its performance is vastly better than it was be fore the re-cell.
HTH Jon N