Whilst it sounds like some sort of review of the PHE methodology is clearly warranted, as it's nonsensical to have the current open-ended system, I would be extremely surprised if there's been a significant over-counting of deaths. We've had fewer than 300,000 confirmed cases, two-thirds of which have tested positive in the last three months.
The likelihood that very many of those will have tested positive, recovered and then died from something else in that timescale is relatively small. I'm sure there will have been some over-counting due to such circumstances, but I'm pretty confident it will have been more than compensated by the under-counting that undoubtedly took place at the beginning when we weren't testing enough people.