Hmmn I see UKMO has the main trough a little further south on Saturday compared to GFS... I guess that explains how the BBC forecast could feature widespread snow and GFS hardly anything.
As for next week, we see GFS very reluctant to split energy in the vicinity of Greenland and send some SE. The result for us is a bizzare soup of very weak disturbances, inadequate for supporting high pressure to our N/NE.
UKMO is not nearly as reluctant - see how the one trough on day 6 becomes two on day 6, one riding up the west coast of Greenland, the other sliding nicely SE to the UK:
Historically, GFS is useless at splitting energy in this way. Well, that's what the likes of Steve Murr will tell you, and from past experience I'm inclined to agree that it does struggle with it.
GFSP had been making a better effort but appears to have gone AWOL for the 12z run. Just waiting for it to turn out that what we're seeing as 'GFS', with it's severe lack of a splitting trough, is in fact GFSP in disguise 
How likely is it to actually split, you may be wondering... well, you need a split in the jet first, and that shows up on all output by day 4 (over far northwestern N. Atlantic), but there are differences in the balance between how strong the right arm of the jet heading across the Atlantic is when compared with the left arm which heads due north toward the west coast of Greenland.
The GFS run has more going north, leaving the Atlantic jet weak - that's why the ridge there is stronger and lasts longer than on the UKMO or GEM 12z op runs.
The GEM run, which is more like UKMO than GFS, though more vigorous with the Atlantic troughs, has the majority of the jet energy heading east across the Atlantic. The results on the GEM run are rather dramatic with a deep trough over S'rn UK - probably very snowy (if viewing, account for the fact that the model modifies maritime-track air masses far too quickly, causing them to be projected as less cold than would actually be the case).
There is room for debate over how decent UKMO is for snow on day 6, but with quite a weak flow and an established cold pool across the UK, I reckon the low-level cold would be sufficient for either snow or freezing rain in the south. Looking further north it's a snowy picture with little evidence of pronounced warm sectors to worry about.
Going forward, something loosely along the lines of days 6-7 of the GEM 12z looks possible, though I doubt it would be that extreme in terms of the high pressure to our NW.
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