It's a horrible lengthy job. You can start with a heat gun: how much you get off easily will depend what's on there. You probably have a few layers of paint over an old coat of varnish or some such. The oldest coats can be very sticky.
That will only get you part of the way. After that, you can either hand-sand or use chemical paint stripper, treating probably just a rod or even half a rod at a time. If you choose the latter route, you'll need to finish with sandpaper or steel wool. It's a long haul.
There was a chap I used for my windows when I first moved in who stripped a friend's staircase up in Mountview Road. He was a carpenter / builder called Arthur Green. He did a first class job, but he wasn't cheap. It was quite a while back though and I would think he's retired by now. So I can't really help on that score.
If you do use someone, be prepared to pay for a craftsman approach or take the time to do it yourself. Otherwise it's better to rough sand out all the drips and just repaint.
thanks hugh... i suspected it would be lengthy or expensive...
I started mine thinking I was fully armed with the resignation to persevere - not a bit of it: I threw in the towel and ended up rough-sanding and repainting the rods. Luckily the rail itself had never been painted.
Probably it would be cheaper and easier to replace the whole bannister with new.