M3 or A316
The road in from the South West, variously termed the Exeter Radial and Basingstoke Radial (though this doesn't actually imply it was supposed to go all the way to Exeter). By the time it was built, with the easternmost section opening in 1974, London's urban motorway proposals had already died, and its terminus was kept far away from the central area to avoid the disruption of another motorway ploughing into the city. But in the 1960s, it was meant to head much further in.
Continues to Basingstoke
R4 Southern Section
R3 Western & Southern Section (or at next junction)
Local connections to Sunbury & Ashford
Local connections to Feltham & Twickenham
Twickenham Bridge over River Thames
Richmond Eastern Bypass (Manor Circus)
Possible local connections to Kew & Mortlake
Chiswick Bridge over River Thames
R2 Western Section
Possible local connections to Chiswick
In from the Basingstoke direction, the M3 would do what it does now, crossing Ringway 4 (now the M25) at present-day junction 2 and heading in towards London. The south-west corner of that interchange might have been bypassed by a short section of M31 - the stage that motorway reached in the planning process is hazy though, and it lies entirely outside the Ringway scheme.
The M3 would meet Ringway 3 either at present-day junction 1, or according to some plans, slightly to the west near Upper Halliford station, where land was reserved for an interchange. From there the route would follow the A316 into London; upgrades carried out in the mid-1970s to Country Way suggest that proposals for a motorway-standard road through here were already well advanced and the junctions and viaducts on this route are likely to be recycled Ringway proposals.
In the late 1960s, Richmond Council approved plans for new roads around the town, including a route for through-traffic built either above or alongside the existing A316, crossing Twickenham Bridge and passing north of the town centre. This new route would have had no junctions between Twickenham and the B353. Manor Circus, the present A316/B353 roundabout, would have been the terminus of a new Richmond and Petersham Bypass, linking the A307 towards Kingston to the new M3 or A316 route.
However, immediately after the plan was approved, a local government reorganisation meant the plans passed to the new London Borough of Richmond upon Thames, which immediately rejected the proposal as "deplorable". This seems to have been the end of any real proposals to upgrade the A316 near Richmond for motorway traffic, though the GLC appear to have remained optimistic about the M3 reaching Chiswick until about 1973. How the route might have appeared between Richmond and Chiswick Bridge is unknown.
On reaching the line of Ringway 2's Western Section at Duke's Meadow in Chiswick, it would continue north-east, terminating at the current A4/A316 junction at the Hogarth Flyover where it would merge into the M4 towards London.
With thanks to Toby Speight for information on this page.