My memory of St Paul’s is a little hazy as it’s been a good few years now since I was there. However from what I remember at least one of the disused lift shafts has a manhole in the bottom, leading down via a ladder to a number of further levels, which are encased in thick concrete. This I presume was the CEGB control room. Leading off from the original base of a lift shaft was a blind heading (presumably leading to the running tunnels), and at right-angles to this was another smaller blind heading leading to a solid wall, with evidence that cables once passed through. As I say, it’s been a few years so I may have his slightly mixed up, but the general gist of the above is sound.
Not sure about Westminster. There’s certainly nothing in the Jubilee Line tunnels. The only features of note are the intervention shaft at Storey’s Gate, vent shafts at both ends of Westminster platforms, and further towards Waterloo a feature on the left-side of the EB tunnel which I believe to be the remnants of the construction working site at Jubilee Gardens, which was reached via a lengthy temporary heading which I believe was backfilled when no longer required. Certainly there’s no way into it if it is still there. LU tunnels are littered with such construction remnants if one knows exactly where to look and what to look for!
Correct about the deep shelters. The access points can be seen in many cases, but completely sealed up. Camden Town is probably the most conspicuous.