Can I get a multi-entry Schengen visa if travelling to two different countries on two separate trips?

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.