High sensitivity to the exact positioning and orientation of the high pressure relative to the UK in the coming week or two.
The initial airmass is clear enough that cold nights and fog are dominating the scene until Saturday.
This weekend, some milder air looks to scoot across, taking effect by Sunday when double digit maximums may occur widely.
Next week, signs are, we see the high centre more or less over the UK in the early to mid-stages, so the inversion may redevelop, depending on how much moisture-laden Atlantic air has been entrained during the weekend. Most model runs shrug much of that off and some draw in some drier air from the near continent, so there seems to be a good shot at developing a new inversion.
Later in the week, there are signs the high will shift west a bit, at least for a time, giving a chance of cold air from the north. Probably just a glancing blow if anything, but it should tend to help keep skies more clear than grey.
Beyond that, the models are keen on repeating the sequence - high pressure spilling back across the UK, then heading west again. An astonishingly 'stuck cycle', which likely has a lot to do with anomalous tropical convection persisting over the western-central Pacific for a record-long spell, by far, for the time of year. A huge distortion to the zonal flow has gradually built up, almost like debris behind a dam - it's going to take some shifting!