It sounds like you have some great experience. My advice would be to look at the person spec on the job description and think about specific examples that show you have the skills or experience required. Many people (including myself) won't mind where you gained that experience, as long as you can clearly show that it's relevant for the job.
You can also highlight the additional benefits that international experience give you, such as a good understanding of other cultures and cultural sensitivities, speaking multiple languages, and an ability to work with people from lots of different backgrounds.
If you don't have any UK experience, it might be worth mentioning in a covering letter or CV that you live in the UK and have the right to work here, to avoid hiring managers making any assumptions that you are still living internationally.