score:2
There is no automatic ban.
However, you will be asked questions about prior visas, both Schengen and non-Schengen. You must answer truthfully that you overstayed in the UK and were subsequently refused a UK visa. Yes, this will decrease your chance of getting a Schengen visa. But lying and being discovered will nullify your chance of a Schengen visa.
The best course of action might be not to ask for a visa right now. If in 10 years time you have to explain one refused visa, it might be seen as an one-off incident, but if you have to explain two refused visas for both the UK and Schengen, it starts to look like a pattern.