Looks like suitable tunnel boring machines are just becoming available in the early 1960s, using pressurised slurry on the face to stop the ground collapsing. This would be a very big and early attempt at it, but it isn't impossible. See
this website.
Very little is impossible, but putting my tunnel engineer hat on this looks incredibly ill-advised for 60s tech. Hell it's not a great route for modern machines.
You can see how awful the geology is on that East Solent section. Starting at Ryde you've got Marl, decent tunnelling rock (most of the Channel Tunnel is in marl) but it is rock. Then you hit the yellow Barton/Becton sands, cutting edge 60s Slurry TBM can probably cope with that (fair bit of R&D cost to develop it though). Then the blue Clays, thick, fissured over-consolidated they are way outside the suitability for even a modern Slurry machine, you are well into EPB (Earth Pressure Balance) realm which is 1970s tech. Small sand lenses 'bursting out' also sound a challenge. Then at the end those interlaced green layers of clay and greensand, just to add to the mixed-face problem.
On their own each layer could be coped with, the Blue Clays would probably be fine with an open shield for instance (BGS has it as fine grained cohesive), but it is the combination variation that is the killer. The tunnel has to do rock, sand then clay, then mixed face all with the same machine and all under-water. If you have a modern variable density mixed mode machine you can probably cope with that, but for a first of it's kind slurry TBM it's getting fairly close to impossible.