Our government agency has tested for the Omikron question. Omikron (paper in English) is recognized just as well by the lateral flow tests as Delta: here.
As for the lateral flow tests, the university hospital of Utrecht has conducted a study. The results: if you have symptoms, the tests work very well (90%). If you do not have any symptoms, they are only about 20% reliable. The good news is: in those cases, people probably have a low viral load - so the question is: is this really a bad surprise as you imply? You could also conclude that you are probably not (very) contagious at that point in time. Here's a link in Dutch. Preprint of the paper here.
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