My partner is awaiting a decision on his UK visit visa. If it's granted, is it possible for him to apply for a Schengen visa once he arrives in the U.K.? I would like us to go to Barcelona or maybe Paris or Rome for a few days (say 3 or 4 max)
He would have to argue that it is undue hardship to apply at the consulate which covers his ordinary place of residence. The premise for the trip would have to be unexpected and sudden. If you try to abuse this, the visa may be rejected, and in any way explaining the special case might be more of a hassle than going through proper processes now.
Follow-Up: The EC writes
Example: A Chinese professor has travelled to London to teach at a university summer school. During her stay, her father, who lives in France, falls seriously ill and in order to travel to France the Chinese woman applies for a visa at the French consulate in London.
The French consulate in London should deal with the application because it would be excessive to require the person concerned to return to her country of residence to apply for the visa.
Example: A Moroccan national who spends his holidays in Montreal (Canada) wishes to apply for a visa to travel to Germany at the German consulate in Montreal. He claims that the waiting time for obtaining an appointment for submitting the application at the German consulate in Rabat (Morocco) is too long.
The German consulate in Montreal should not accept to deal with the application, because the justification is unfounded.