Aerofoil1
The Cessna 172 is a good place to good place to start the search but I would not go buying a Cessna 100 series aircraft until we see how this SID,s thing pans out.
I take it from the post that you made you need to have the four seats full, this means for all practical purposes you are unlikly to be able to go very far with any less that a 180 HP engine, we can discount Permit types as only the RV10 is able to for fill this requirement and unless you build one yourself you are unlikely to find one on this side of the pond.
To keep the cost down it would be best to seek a common type such as a PA28, both the PA28-180 & 181 would fit the bill with four people they can lift enough fuel to take you half way down France from Oxford. Parts are cheap and supply is good, the aircraft are known by most maintenance company's so you should have few surprises. The older 180's are getting a bit long in the tooth but if well maintained this should not put you off.
The TB10 is another place to look, there are some good aircraft to be had and it will match the PA28 in terms of performance and cost.
I know that some on this forum will suggest some other aircraft, most would fit the bill but the issue of parts availability and cost will be a factor.
If you can find a hangar to keep it in the real star of this sector of the market is the Robin DR400 it will do all that the PA28 or TB10 will do but 10-15 kt faster and using about 30% less runway, the performance is so good that the 160 HP Robin will match the performance of the other 180 HP aircraft.
With two adults on board the DR400-180 (fitted with suplimental fuel tank) will fly Oxford to Bilbao non stop and land with two hours fuel in the tanks, the only snag is that when the supplemental tank is filled the C of G moves aft and effectively turns the aircraft into a two seater unless the rear passengers are very light.