The dual passport holders in our family have no problems with what you are proposing. Most countries require you to enter or leave using that country's passport if you hold one (i.e. leaving or entering the EU requires you use your EU passport, the same for the US and, I assume, the UK). And you should, in general, enter and leave a single country (or block like the EU) using the same passport. For entering a third country, I think you can use whichever works best for you and so that passport that should be presented at check-in.