What are the issues involved in converting a quadjet into a twinjet?
Yes, you'll get a wing redesign to allow for the higher weight and thrust of the engines: the inboard engine mounts now need to be twice as strong.

You can also run into ground clearance issues. An engine twice as powerful is often also larger. Here's a GE 9X engine being tested on a Boeing 747. You can see it's pretty close to the ground, and needs a really short pylon.



Going from 4 to 2 engines also means replacing half the instruments in the cockpit. You could save a lot of weight by removing the wiring for the outer engines, but that'd be a major operation. 

Historically, the advent of high-bypass engines coincided with the introduction of widebody aircraft for a huge increase in passenger capacity and efficiency. So e.g. Boeing could make more money by offering a new design instead of selling 2-engined 707s. 

For the remaining 4-engined airliners (B747 and A380) no engine is powerful enough to allow a 2-engine conversion. An engine powerful enough would be even bigger than the GE 9x shown above, and would scrape the ground when installed on a 747. 

Finally, I'll leave you with a response to footnote 1: The VC-10 was used to trial the RB.211 engine, which was 2x as powerful as the original Conways: