Contact tracing was launched on 28th May [2].
It seems to me that what went wrong was a large number of UK residents were overseas in February/March (sspecially on ski breaks) to Italy, France and other mainland Europe destinations. The Government didn't impose any emergency isolation or quarantine on those arriving, as was the case for those arriving on repatriation flights from Hubei and other Chinese destinations.
Given much of the scientific analysis of the virus points to strains being descended from those in mainland Europe, it seems a key reason the UK wasn't "head and shoulders" ahead was the scale of this international leisure travel to countries that ended up most affected. Had those returning been put into quarantine, there might have been a very different outcome, but it's hard to say for sure.
[1] https://www.bbc.co.uk/news/uk-52774854
[2] https://assets.publishing.service.gov.uk/government/uploads/...