If you're travelling from Cork then the Rosslare to Fishguard crossing is your nearest sea route and this brings you to Pembrokeshire, a beautiful part of South Wales. Look up Tenby and its surrounding villages, St. Davids is the UK's smallest city, it has a medieval cathedral. From there you can head inland to the Brecon Beacons and then north through some of the Uk's best biking country. Visit the Elan Valley, the reservoirs supply water to Birmingham, 100 miles away, and were begun by the Victorians. Snowdonia is well worth a visit for the scenery and there are several typically British seaside resorts you can take a look at (Aberystwyth, Barmouth, Porthmadoc, Llandudno among the larger ones). Castles are everywhere if thats your thing. Or preserved narrow gauge steam railways, Wales is full of those as well! If you see a sign anywhere saying "Narrow Mountain Road to....." then if the sun is shining, follow it and be rewarded by some stunning scenery. (Rhayader to Aberystwyth, Macynlleth to Llanidloes are two of my favourites). Campsites are plentiful all around Wales.
Coming into England, I live in the Midlands and can recommend many places that might not feature in the tourist guides. The counties of Shropshire, Herefordshire and Worcestershire are all great for bikers. The Black and White village trail on the Welsh borders, Ludlow is well known as a "foodie favourite", if Industrial Heritage interests you then see Ironbridge Gorge and its many museums dedicated to the birth of the Industrial Revolution, the Long Mynd can give you a rare sense of remoteness in mid-western England, and the county towns of Shrewsbury, Hereford and Worcester are large enough to be worth a visit, small enough to do on a flying visit by motorcycle.
Continue south-east into the Cotswolds, very Olde Englishe but also very touristy in the summer. Stonehenge isn't that far away from here.
Devon and Cornwall will be well covered in the tour guides but are VERY busy in summer, as are most of the south coast resorts. The east coast area around Great Yarmouth has given us some nice family holidays in the past but never gave me the urge to go back on the bike.
Heading North again, Keith has already mentioned the Peak District and as he's local I'll that to him! I'll also second what he says about Yorkshire, but will add its worth continuing north into Northumberland on your way to Scotland, keep off the M6 motorway, motorways are boring by bike anyway.
No bike tour of England is complete without a visit to Cumbria and the Lake District, theres no doubt that biking is the best way to tour there as again it is extremely busy with tourists.
I've only ever visited Scotland with the family but it is well and truly on the list for bike trip. PadG's write up is excellent.
You can probably guess that I'm a regular visitor to Wales and love the place. In fact, had a ride-out with the lads to the coast at Barmouth yesterday in some very unseasonal sunny weather. Gales again today, though!
Consider membership of the Camping and Caravanning Club (Welcome to The Club - The Camping and Caravanning Club
), they have a UK wide network of sites, in all sorts of areas from the very remote to near London where you can use the train to get into the city. I'm no longer a member but always found club sites to have clean and well maintained facilities.
I'll be more than happy to assist you further if I can, have a great trip,