The reason people can get away with Copenhagen is that decoherence is absurdly effective in all realistic cases, causing there to very little uncertainty about whether and when a measurement happened. Insofar as we can all agree when a measurement happened, physicists operating under Copenhagen are going to be just as effective as those who understand MWI. And for most of the theoretical and especially computational aspects of quantum computers, this is the case. The only place where someone operating under the Copenhagen interpretation is going to falter is when they have to think hard and in novel ways about how to prevent decoherence.

So I really don't think interpretation is going to be correlated with advances in the computational aspects (i.e., what kind of problems are quantum computers good for once they are built, etc.) but I would predict that future insights in how to build computers which are robust against decoherence will be dominates by those who understand MWI.