Today, it is easy to detect cosmic rays with a home-made cloud chamber. Hence, Wilson should have detected them with his chamber, right?
Wilson started to develop the Cloud Chamber in the late 19th to reproduce the optical glory. He soon recognized it could be used to detect particles. In 1911 he perfected his device to take pictures and started tests with X-rays and beta particles.
During his experiments there were some unwanted ion productions in the chamber even with no source of radiation. Wilson suggested that that may be due to extreme radiation coming through the atmosphere, what would be the first idea of cosmic rays. He proceeded with tests in "isolated environments" such as tunnels and the effected did not change so he abandoned this idea.
In Wilson's Nobel Lecture he does not mention about cosmic rays so it might be possible he worked only with X-rays, alpha and beta particles at least in the early days of the Cloud Chamber.
The Austrian Nobel price winner (1936) Viktor F. Hess discovered the cosmic radiation during several ballon flights in 1912 with specially designed ionization chambers. He discovered that (unexpectedly) the penetrating radiation increased with altitude and concluded that there must be a extraterrestrial source for this radiation.(Hess paper 2012) Wilson's cloud chamber made the tracks of ionized particles visible. He did not use his chamber to study cosmic radiation which was unknown to him in 2011. The cloud chamber detects ionized particles which are mainly produced by terrestrial radioactive elements at sea level.