I didn't mention Zinsser BIN,
@Canyer did. My comment about BIN being a modified white polish were aimed at that.
However, it still doesn't answer the question I asked yesterday - namely "Would either of these be any use to overpaint the existing undercoated sills or is it still a 'strip back to the wood' job?"
To which I'll quote a previous post of mine:
I suspect at this point your daughter will also need to scrape back all the paint then try again - but I would suggest trying a small area
The fact that the original finish is bleeding through the applied paint indicates that the new paint needs to come off before you do anything else because the finishing regime just isn't working. To my mind there is no point in trying to seal the current mess in because that absolutely
NEVER works (a bit like continuing dig a hole?). If the new paint is fairly recent it may just scrape off, or it may need additional help from a heat gun. Either way there will be less pain if it comes off
As to the original finish I stated that
if it is an oil finish (as appears likely because you stated that the old finish was bleeding through), you need to remove as much of it as you can with a solvent first
after scraping the new paint off, and after that you can seal it with white French polish (or for that matter Zinsser BIN):
Before painting you normally try to remove as possible of an oil finish using an appropriate solvent (finding the right one can take some trial and error - white spirits, methylated spirits, naptha, etc are all possibles) on a well washed white cotton rag (old T-shirt, etc), then seal it (again with a coat or two of French polish). Shellac in French polish is very good at stopping incompatible finishes affecting each other as well as sealing stuff like this - but you really do need to knock back the oil a bit before applying it.
So I feel I have already answered your questions and to recap:
1. You need to get the new paint scraped off (and obviously to clean up any roughness with sandpaper)
2. You then need to degrease the existing finish and remove as much oil 'contamination' as possible.
3. (Additional, as no information was forthcoming before) If the finish is a glossy lacquer finish, it needs to be lightly sanded to provide a mechanical "key" to the surface (standard procedure for gloss painting to prevent peeling of applied coats - generally not required when going over oil finishes). In view of the applied paint this might not be a bad idea anyway. This is not a strip back to the wood sanding, just a light sanding with a 180 to 220 grit sandpaper to mechanically key the surface
4. Because of cross contamination between finishing materials a barrier layer of white French polish (or for that matter Zinsser BIN) needs to be applied and allowed to dry thoroughly before priming and glossing the woodwork
5. You can now prime the woodwork again...
Apologies if this wasn't stated clearly enough before