From the video and most of the lyrics, the track is about a Morris Minor 1000 convertible. The lyrics, however, do lend themselves to a little artistic license, e.g. the Morris was not built by the Tyne, it was built at Cowley, Oxfordshire. This is now the manufacturing plant of the BMW owned Mini.
When this track was written, the A45 ran from Felixstowe to Birmingham passing through Northampton. That road has largely been renamed to the A14 from Felixstowe to Cambridge and the A428 from Cambridge to Northampton.
The b-side to the single was "Animal Farm", a mostly instrumental reworking of the song "Tomorrow's Dream". The 12" release of the single included the song "Riding on My Bike", which is basically a rewording of the main track, sung by Lee Thompson.
"Driving In My Car" was the thirteenth in a run of twenty consecutive UK top twenty hits for Madness, and is the only one of those twenty never to have been officially released in the USA in any form.
The video shows Madness as car mechanics larking about in their workshop, and in their normal suits driving around in their "Maddiemobile". The members of fellow ska/pop group Fun Boy Three make a brief appearance, trying (and failing) to hitch a ride to their home town of Coventry, which the A45 road mentioned in the song passes through.