Do you have a particular power supply in mind already? If it is stabilized/regulated, it should work just fine as you are right in your last sentence. An unstabilized, too "heavy" power supply might be a problem as they will output a higher voltage with a smaller load.
If you want to be sure, you can measure the output of the power supply with nothing connected to it. It should be at 6V dead center, not eight or nine or something...
Have you simply tried to get an original replacement power supply? Through a quick googling US versions seem very cheap even in the official Slingbox store, there must be UK versions available as well somewhere.