Newbie question about BMS boards and 18650 cells

I would like to replace/convert some old (non working) 18v Nicad power tool batteries to use 18650s instead. I’ve got some brand new 18650s and some 5s battery management system boards, but as a non-engineer, there is just one piece that isn’t in the wiring diagrams and seems pretty critical.

Based on my online reading, it appears that 5 cells will give me the best match for 18v tools, it sounds like 4 isn’t enough

I understand that each 18650 need to be charged at 4.2v, and if I was just charging one, that makes sense. When I look at the diagrams for the BMS wiring, they don’t indicate line input voltage for five 18650s wired in series along with the BMS. Does my input voltage need to be 5 x 4.2 (21v) or do I have input of 4.2 to charge the whole system, and still expect it to deliver ~21 on the way out? Sorry for such a basic question.

Here is some additional info; I have two chargers that are interchangable for the old batteries; they indicate output of
21.75v 85ma
24v 210ma

In an ideal world, I’d use the existing chargers against the new 18650 5-packs to charge, and then just plug into my 18v tools and be able to use them again.

Thank you for any clarification!