Are you looking to replace the balustrade rail (the top part) or the spindles?
There’s plenty on each subject in YouTube, if you search.
As far as the’ period correct’ profile of balustrades is concerned, it’s best to find a similar property (in the same street?) and try to copy what they have…… and what you are happy with. The basic shape didn’t alter much – after all, the human hand remains the same, so there’s lots of latitude for choice.
When the balustrades were made by hand, there were a number of specialised tools that are hard to find nowadays, though you can replicate most of that with modern (power, if you need to) tools.
Spindles are also a basic item. If they are all missing, you can buy new ones, machine made, if you need to; most builders merchants can supply them. If some are missing/broken and are a particular shape, you may get some made to suit and match. Any competent wood-turner should be able to do this for a few beer-tokens. You will need to remove a sample in good order for him to copy.
As a final note, (I don’t know where you are) but in the UK, there are statutory building regulations concerning the construction and repairs to stairs and the minimum distance that spindles may be set apart.
Worth some research to get it right…..