The War Machines has a mix of some very prominent and rather obscure locations. The big novelty in this story is the Post Office Tower, now BT Tower. It was the first full contemporary Earth story paving the way for the likes of the Faceless Ones, Web of Fear and eventually the whole UNIT era. Hartnell works wonderfully in this story and it's a shame that this came late in his era, leaving his Doctor often in historical adventures or on alien worlds. But this isn't a review, just a brief look at some of the locations used in that story.
Immediately jumping out is Bedford Square only a minute's walk from the British Museum. Here the TARDIS lands in the first episode and is shot from various angles including a window of one house. In the row of photos to the back of me when taking this photo is No. 41 which doubles for the Royal Scientific Club.
From here the Doctor heads for the Post Office Tower which can be seen roughly to the North, however he is seen in the next shot approaching from Fitzroy Square along Conway Street which is just to the north of the tower.
The Tower itself is largely unchanged as seen in this shot from Cleveland Mews just off from Maple Street. The angle achieved in the story must be from a platform or window above the ground floor. The Tower has been closed to the public for a long time and has a rather permanent plaque on the wall stating this. :-(