This information is from an archaeologist at The Kotel.
The beginning of regular prayers along The Kotel instead of on Har HaBayit (the Temple Mount) apparently starts 360 CE (Byzantium)
There were times like 9 B’Av where we ‘paid’ (read bribed) the Byzantines to let us pray on Har HaBayit.
Between 700 – 1260 prayer was along The Kotel in various locations but primarily inside Warren’s Gate and Mul Kodesh Kedoshim.
The area that is today known as Robinson’s Arch had a cemetery and was also used for prayer most probably during this same time period. This is where the engravings on the wall under Robinson’s Arch are from.
From 1320 – 1338 the Mamaluks began expanding the Old City by building along / attached to what most people call the Kotel.
Up until 1546/7 there was very limited prayer by the Jews along the Kotel because access was so restricted. In this year there was a large earthquake in Jerusalem and a number of the Mamaluk building attached to The Kotel crumbled. In particular in the section that has become know as the Kotel; the main prayer area used today.