In general you have to apply for a visa from your country of citizenship or residence. You are NOT a UK resident, technically or otherwise. Unless someone else knows any different, your best bet is to call up the Schengen countries' embassies in the UK and ask each of them individually whether they will allow a tourist in the UK to apply for a visa. Even if only one of them says yes, you can more or less visit any of the countries as long as the issuing country is the "main" place you visit.

The other thing to check is, does your UK visa say MULT on it? If not, you won't be able to re-enter the UK even if you did manage to get a Schengen visa.