Rome is famously built on quite a few layers of ruins itself, so I don't think you can draw that particular comparison.

While some parts of the ancient city were not excessively buried (the Colosseum may have been only a few meters below ground when it was excavated), many of the famous sites in Rome appear to be at modern ground level only because they have been extensively excavated. The Forum, for example, was significantly buried. An interesting relic of this can be seen in the church of San Lorenzo in Miranda on the edge of the Forum, which was built out of the Temple of Antoninus and Faustina. While currently excavated to something like a classical-Rome ground level, you can see the church door and the bases of pilasters built at a previous ground level.

Other churches, being continuously in use, provide some wonderful illustration of the stratification of Rome. If you take the Scavi tour of St Peter's Basilica, you can see the St Peter's Crypt, which is the ground level of the previous basilica, and below that the Roman necropolis which was ground level in the 1st or 2nd century. The Basilica of San Clemente al Laterano similarly allows you to tour, with rather generous ceiling heights, the present church (which is several steps below modern street level), its basement (which is the previous basilica), and earlier residential structures beneath that.

You don't have to look very hard to see other signs. The "sunken courtyard" around Trajan's column is simply an excavated area, showing the difference between modern and ancient surfaces. I particularly like the colonnade transitioning into the unexcavated area, as a little taste of what Renaissance Rome would have looked like.

One thing that Roman ruins did have going for them in relation to London is size; many Roman structures survived at least partially above ground and were reused or left alone instead of forgotten belowground and built over. Structures in London were less grand and less likely to be stone, thus turning into shallower layers. You can, however, still find parts of London's Roman wall still standing, as it was large and stone.

Build up of land happens due to many factors. Vegetation left to grow, especially in warmer climates, can create layers of organic debris with surprising speed. Historically, habitation also tended to create landfill, which was a major factor in depositions in Rome. This takes many forms: collapse or demolition of buildings, intentional dumping, everyday debris, etc. Low lying areas may silt up. Events such as flooding or earthquakes or volcanos can also bury things. These factors are all very local and will differ by site.