Mobil's plan was built on an idea initiated by the Department for Transport. BP were hoping that by coming up with their own idea at Gaydon, the Department would be forced to go for the cheaper option where most the administrative work had been done by BP for them. This plan backfired, as the site at Warwick was seen as being better in most respects, not least because it didn't pass traffic through a low-capacity junction. Even so, it was one of a few clashes which eventually saw private companies given full responsibility to develop service area plans.
Long before the M40 was built, there are references to service areas at Gaydon and Chesterton being high priorities to open. This is unlikely to be the same site, but the other plan is difficult to uncover.