Underground is not possible there due to the amount of underground infrastructure dating back hundreds of years. Look at the upheaval at adding a new tube station and the cost and mayhem that caused (can't remember where the last station added was but I seem to recall a new one on the Central Line last time I was in London a long long long time ago).

Underground road tunnels are certainly possible. London's clay is over 400 feet (120m) deep (deeper as you go east, shallower as you go west) leaving plenty of easily-tunneled space if you go below most of the existing infrastructure.

There is currently a "super sewer" tunnelling project (Thames Tideway Tunnel) which goes down to a depth of about 75m, below other infrastructure.

The Elizabeth line and other tube lines are shallower, for access reasons and because building stations at a great depth is more expensive. But a road tunnel could go deeper, especially if it only permitted zero-emissions vehicles so that ventilation is easier.

Of course, it'd all be very expensive. And there is an argument that building more road relief capacity is not a desirable thing as it just encourages traffic elsewhere.

This was tried with varying levels of success over the past 150 years and is known as the tube.

Ah yeah, the sky tube. Funny how you all focused on the "maybe underground" part.