Doesn't meet your max 30 minute walk - see below for walk - but I would have suggested Newport Pagnell services on the M1 but the on-site Ramada has recently been closed at short notice to house asylum seekers.
The walk would be from MKC station up through the city centre and down through Campbell Park to the canal. Left along the towpath to Great Linford and then follow the old Newport-Wolverton railway line mentioned below, now one of MK's redways, until you pass under the M1 bridge. Then turn left and take the path to cross Wolverton Road, turn right, then left at the Red House (now called something Smoke Shack or suchlike) and up Linford Lane to the M1 services.
That way you would have walked along part of an old, repurposed, railway line to get from railway to motorway
Now if we were living pre September 1964 you could have got a train to Wolverton on the WCML and then taken the "Nobby Newport" down the branch line to Newport Pagnell station. The walk from there to the M1 (opened late 1950s) would have been within your time limit. Or you could have got off at Great Linford, which would have been a slightly quicker walk.