Generally speaking, that's not how it's supposed to work and there are many reasons why each consulate might want to avoid giving a visa for the other trip you are planning, see Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa? and Applying for a Schengen visa when there is a gap between trips to main destination and country of entry for more details.
That said, as a UK resident, you might get lucky. I have heard about some folks who got a one-year multiple-entry visa the first time around without even asking for it or planning another trip. It does make sense: Someone who resides in the UK has made it through the rather arduous process of getting a UK visa, can have many opportunities to go mainland Europe but little incentive to stay there and earn money illegally. All this makes you a low-risk applicant and the consulate could save everybody some effort/money by giving you a multiple entry visa.
Finally note that there is a third type of visa that would seem to fit your itinerary, namely the two-entry visa. In many respects it's closer to a single-entry visa than to a multiple-entry visa. It was intended to make it possible for foreign tourists (but not necessarily residents) to add a stay in British isles in the middle of a tour of Europe without having to worry about requesting two visas on the road.