Yes, it can be done in 1 day.
I have not driven the English leg that I suggest here but my route avoids the built up conurbations around Leeds & Manchester - which you would pass on the busy M62. My route uses an ordinary 'A' road (non divided) which is north of these big cities and mainly passes through countryside.
Leave Thirsk on A168. Go straight across M1 Junction 49 and follow *minor road to RIPON. Then, A61 towards HARROGATE but take B6161 at Killinghall to join A59 going west towards SKIPTON. Follow the A59 all the way across to the west side of Britain & join the M6 at J31 (just East of Preston) & then head south. (Not having driven this route, I can't be 100% sure it is the best).
Anyway, whichever way you go, you will eventually leave the M6 & take the M56 towards CHESTER = well worth a visit. Then, it is the A55 North Wales Expressway - which I suggest you leave at J19 (Llandudno Junction) & follow A470 south to LLANRWST. Here you will see an old stone bridge crossing the river - leave the A470 & cross this humped bridge to join the B5106 on the other side of the valley. Now turn left (south). You will eventually reach the A5 just W of Bettws-y-Coed.Then turn R & take A5 westbound to CAPEL CURIG > A4086 >A498 to BEDDGELERT & finally through the Aberglasyn Pass to PORTHMADOG. (This route from Llandudno Junction is incredibly scenic but not exactly fast. Your alternative is to stay on the A55 North Wales Expressway to Bangor & then the A487 via Caernarvon to Porthmadog. The roads are better but it is a longer route & lacks the stunning scenery of the first suggestion).
Try and follow these routes on Google Earth to help you decide what you think will suit you best. * This junction is complicated so make sure you draw a little sketch map from Google Earth to take with you - should you choose this option. Another route from Thirsk to Ripon exists just to the north.