Firstly most of the London Underground was built cut-and-cover which is no longer practical in modern cities. Secondly, this will be much smaller than even the Tube. Crossrail’s tunnels will be about 6 meters, most existing are more like 4. BoringCo is talking about something like 2 meters. Note that a 2m tunnel requires excavating 4x less earth than a 4m tunnel and 9x less than a 6m tunnel.
Most modern tunnels built for existing transit systems are built to fit existing rolling stock, so the idea of smaller tunnels gets ruled out early on. In the past, smaller tunnels were used. In fact, one of the most extensive small tunnel transit networks was in Chicago: https://en.m.wikipedia.org/wiki/Chicago_Tunnel_Company
Of that underground, I Elise the majority is the deep tube bored lines (pic, jubilee, bakerloo, central, northern) rather than pretty much just the circle that's cut and cover.
Most modern tunnels built for existing transit systems are built to fit existing rolling stock, so the idea of smaller tunnels gets ruled out early on. In the past, smaller tunnels were used. In fact, one of the most extensive small tunnel transit networks was in Chicago: https://en.m.wikipedia.org/wiki/Chicago_Tunnel_Company