There are (for England) three different sets of deaths figures. Only 2 for Wales, Scotland and NI I believe.
The deaths within 28 days set is an easy one to produce, practically. So that one comes out first, and it gives a rough idea what's going on, but obviously it's not accurate (and I suspect it's less so now than it used to be as we now test pretty much everyone in hospital, and quite a lot of people die in hospital, from all sorts of things).
There's also a set for deaths within 60 days, which again is easy to produce but I think has been found to tend to overstate deaths from Covid. This is only available for England I believe.
The ONS then use death certificates to find (near as possible) the number of deaths actually caused by Covid. This is by far the most accurate figure. However, it has a considerable reporting lag as they have to wait for death certificates to be issued and so on.
So what happens is that the press, government, etc generally use the less accurate figure as it's more up to date.