Right, from London, I'd imagine that there are two ways that you'd probably want to consider:
(1) M40, M42, M6, M54, A5, A55 - which should hopefully be pretty clear up until at least Shrewsbury (as you'll be on motorway and dual carriageway all the way); after Shrewsbury it's mainly single carriageway until you get to near Bangor and join the A55 - which is then dual carriageway all the way to Holyhead from there. That route does cut across Snowdonia though - through Betws-y-Coed and Capel Curig - so you'd want to double check to make sure that stretch of the journey was OK.
(2) M40, M42, M6, M56, M53, A55 - this is a longer route as you're not cutting a corner off by going via Shropshire - but are instead going all the way up to near Warrington and then going via Cheshire and along the North Wales coast. However it's motorway or dual carriageway all the way.
I'd be inclined to check the AA or RAC route planner websites and get an opinion from there as to how long it's likely to take - and then make allowances based on the weather...