The hospital deaths announced by the BBC (and others) are just that - those announced that particular day, with the announcement often coming some time after the actual death.
Spreadsheets containing the daily hospital deaths announced for England (only) broken down by date of death are on the NHS England website, and show a different picture.
For example, of the 244 hospital deaths announced today (England only), 60 happened more than a week ago - the oldest being 27th March.
The 627 you refer to was 'all settings' for the whole of the UK - hospital deaths accounted for 374 of these, and again, a significant proportion of those happened at least a week earlier.
Transmission in the general population is now reasonably under control - care homes are another matter.....