Using National Cycle Routes, and starting from Rhyl (based on your screen name!), your best route is probably to hug the North Wales coast as far as Flint. The go via Chester, Runcorn, Widnes, east across to Hadfield and Doncaster, north to Selby, then east via Hull to Withernsea:
http://www.sustrans.org.uk/assets/files/genera l/sustrans-map-2009.pdf
(That map's a 3.5Mb download, so it might take a while to display).
Use the map to find the route numbers, then get more details from the 'related pages' links here:
http://www.sustrans.org.uk/what-we-do/national -cycle-network/route-numbering-system
If those links don't give you distances, you should consult a decent road atlas (or, better, OS maps). As a guide, the AA website gives the distance for the recommended road route (for cars), between Rhyl and Withernsea, as 185 miles. The National Cycle Routes will add additional mileage, so I'll guess at 230 miles.
12 to 15mph is a fairly leisurely cycling speed. I'll use the lower end of that range (partly to take into account your slower average speeds on the hills of the Peak District), so I'd estimate around 20 hours cycling. So, with 6 or 7 hours (fairly leisurely) cycling per day, I'd estimate
3 days.
Chris