All of the places mentioned so far are well worth a visit. However, bear in mind that Yorkshire & the Lake District will be a longer drive to the north & much of this will be on boring expressways - some passing through conurbations such as Birmingham. I would be inclined to take the train out from London to Oxford (or go by bus from Heathrow). Walk around Oxford & then hire a car and drive out to Blenheim Palace (about 7 miles NW). Find a base for about 3 nights in the northern Cotswolds & visit Bourton-on-the-Water, Lower Slaughter, Snowshill, Broadway, Chipping Campden & perhaps Stratford-upon-Avon. (A week in the Cotswolds is too long).
I would then head off west into Wales (www.visitwales.co.uk) which you can easily do by following route A40 to a second base in the Crickhowell/Brecon area for say another 3 nights. http://www.breconbeacons.org
(The M50 is an alternative route as far as Ross-on-Wye). You could from the Brecon area follow route A470 all the way up through the scenic middle of Wales to the Snowdonia National Park. From the Brecon area, you could continue to follow the A40 to SW Wales (Pembrokeshire - consider Tenby). http://www.visitpembrokeshire.com; http://www.visittenby.co.uk
Alternatively, you could drive direct from the Cotswolds to this area looking at interesting places along the way. Here is a reply I gave earlier to someone going from the Cotswolds to NW Wales:>
Leave Chipping Campden & join A44 going towards Evesham - which you loop around on the by-pass. Then B4084 towards Pershore
& pass south of Worcester to rejoin A44 going towards Bromyard. (It is very bendy). As you approach Bromyard, look out for signs directing you R to LOWER BROCKHAMPTON HALL. It is a bit of a drive but at the bottom of the hill you will find the Hall. (Everything you expected Ye Olde England to be).http://www.nationaltrust.org.uk/brockhampton-estate/
Return to A44 & continue W to join A49 heading N. Do not take Ludlow by-pass but drive into this small town and take a look around.
http://www.ludlow.org.uk
Now back onto A49 going N (towards Shrewsbury). About 10 minutes up the road is this:> http://www.english-heritage.org.uk/visit/places/stokesay-castle/
If you are running late - stay on A49 heading for Shrewsbury & then A5 going NW into Wales via Llangollen (as before). However, if you have time, you may wish to fork left just N of Craven Arms & follow A489 to Church Stoke & then A490 towards Welshpool in order to visit Powis Castle. http://www.nationaltrust.org.uk/powis-castle/ (ALTERNATIVE IS TO HEAD WEST TO DOLGELLAU AREA FROM POWIS CASTLE via A458 & A470 or follow route below).
On leaving Powis Castle - take A483 towards Oswestry / Chirk for A5 into Wales via Llangollen (as before).
If you arrive in the Conway area early (unlikely), you could do Bodnant Gardens as it is just off your route. Otherwise - do it on the Saturday. (It is at its best in May - don't miss the Laburnum tunnel - which too the left of the entrance & will be in full bloom in May).
You really have not given yourselves enough time to 'do' north Wales. In addition to Bodnant, I would prioritise driving back to Betws-Y-Coed & then A5 to Capel Curig > A4086 > via Beddgelert & A4085 via Garreg to PORTMEIRION.http://www.portmeirion-village.com
Alternatively, if the weather is clear, you may wish to prioritise going to Llanberis and up Snowdon via the mountain railway. You can then add Caernarvon Castle to the Saturday trip before returning to Conway.
http://www.greatorme.org.uk
You might wish to fly on this zip-wire:> http://www.zipworld.co.uk
PS> When going from Conway to the Lake District, Chester is worth a look.http://www.visitchester.com