either side
There will be no expansion along the length of the strips, so you don't need to have a gap at either end. Wood expands and contracts along the width and depth, not along the length. Be aware that if you wedge in pieces against the stringers to get a tight fit you will be forcing the stringers apart incremently as you go. Very bad thing to do.
I'm not sure that I understand you completely. What size flooring planks do you have that you want to cover the steps with? I also don't know how the nosing is supposed to work. How deep (front to back of tread) is this nosing?
I am concerned that you may be letting yourself in for problems down the road. A stair takes terrible abuse. Having the treads made up from several(?) individual strips with a nosing attached is asking for trouble. Most likely the least troublesome thing will be that the stairs will creak like a son of a bitch from all of those strips moving independently. A bigger issue is what is holding that nosing in place.
In general there's a reason that treads are made from a single piece of wood. Is there any reason, besides the fact that you have some left over pieces that you'd like to use, that you can't use single piece treads and stain/finish it to match your flooring?
R