In reply to mattrm:
I cycled between them in 2012, but at a leisurely pace, as part of the British 3000's- South to North
Staying in B&Bs tends to cut into your routine because of late breakfasts.
Of the people I've heard of who have done it, most have had support vehicles. I guess that eases such issues, being able to set up and break camp and drum up breakfast from the vehicle.
Scafell was approached from Carnforth via Windermere and Keswick. Carlisle by pleasant back roads east of Skiddaw. There's a NCR running alongside much of the A74 which avoids dicing with heavy traffic to Gretna and North.
The route through the string of Lancashire towns will keep you on the alert- roadsigns, parked cars, traffic lights and junctions.
The Cycle route from Chester, along the N wales coast is good and flat except West of Aber, where it heads off uphill. The A55 pavement disappears at this point too, so the only option is the A55 itself. Not recommended.
I'd look closely at the maps from here to Llanberis, and plan well ahead.
Just before I set off from home, I met a man, Clive Hartfall, Who did the three peaks by bike, from his home in ?Waunfawr as a way of celebrating his 60th birthday. IIRC it was a pretty quick time too!