This is factually accurate, but as we usually test for COVID-19 in people that we are suspicious of having the disease (as opposed to a random sample a la ONS) the prevalence is quite substantially higher than the population prevalence
That is incorrect. With a % positive of 0.5% (the average over summer). and a specificity of 99.95% you would expect that roughly one in 10 cases detected would be a false positive, so around 10%
As I've noted, the specificity is probably higher, but that's the lower bound 
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99.95% for ONS. 99.6% for pillar 1 and 99.2% for pillar 2.
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Those are the ABSOLUTE MINIMUM values, yes. If every single positive was false (extraordinarily unlikely) in testing datasets, those are the lowest possible values for specificity. A more realistic range would use that as the lowest estimate and 100% as the highest
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