There is a lot of nonsense being written about freight through London.
I live right next to the south London Line, which is the main freight route from the Channel Tunnel, Kent and points on the south side of the Thames estuary for trains that need to get north or west of London from those places. Despite various schemes over the years to have a 'bypass' line (eg via Redhill and Guildford) for such traffic none has come to fruition and therefore all of that freight traffic (now with the exception of a small proportion that goes via HS1) is routed through relatively central south London through densely built up residential areas. If there's a conspiracy to send freight elsewhere it's not been very successful. This traffic mainly heads up the WLL after reaching the Clapham area.
Heavy aggregates trains pass regularly. Freight is mainly slotted in between rush hours when the line is most intensively used. Chunnel traffic (although there's not a lot of it at present) tends to be taken through the tunnel overnight, so trains from there often pass in the wee small hours. I'm not aware of any particular directive that freight should not run at night so as not to disturb people. If and when Chunnel traffic picks up again, I expect a lot of it will fill the currently unused paths during the night because there's not loads of spare capacity during the day.
Regarding being shaken out of bed - the house does shake, sometimes with things visibly moving. It depends on things like how much water is in the ground and I assume the weight of trains and whether the speed they happen to be going at hits some kind of resonant frequency. What's noticeable though, and commented upon by most guests, is that there's more disturbance from road traffic on a not especially major street. And the house also shakes when an HGV goes over the speed bump just up the road.
Freight going by road disturbs (and indeed kills) far more people than that going by rail.
And if like me you decide to live next to a railway you can't expect it never to make any noise or disturbance.
The most fun is when the tamping machine or rail grinder comes past repeatedly at 3am!