Right now in the UK, the basic testing infrastructure is more-or-less world class as far as I can tell, with a level of testing that's basically unprecedented outside of the US (though some other European countries are catching up over the last few weeks) - it's still having trouble coping right now, but that seems to be the case throughout Europe and elsewhere too. That is very much not the impression you'd get from the media. They keep pushing stupid narratives like the UK needing South Korea-style mass testing when in reality the UK is testing about an order of magnitude more people for Covid, and South Korea charges most people with symptoms for testing in order to discourage them from getting tested and using up the limited capacity whilst the UK tests for free.

Also, the government also seems to be working on the testing infrastructure - from what I can tell, the problem is that as it turns out that things like expanding testing capacity has a lot of lead time, and the capacity expansion is a couple of weeks later than it ideally should have been.