It is worth remembering that a great many species of plants growing in the UK are not native ones - in fact there are at least 2000 'invasive' species, most of which have adapted to the climate and existing plants and animals perfectly well. However, when we speak about invasive species it usually means the ones that are causing problems by growing too fast and/or taking over, even killing the existing ones.
One of the best known of these is Japanese knotweed, which will grow almost anywhere and is very hard to get rid of. Like other weeds it should never be composted but treated as waste. This is the best known, but in fact there are problems with other types of knotweed too, such as the Himalayan.
Some invasive 'weeds' are also much loved plants in the UK, such as the rhododendron. Some consider this a pest because it takes over, despite its beautiful flowers.
You can see a list of invasive plants here.