How did the cryptographers of Bletchley Park figure out the chi stream of the Lorenz cipher?
The answer can be found here:

The initial function of Colossus was to help determine the starting
point of the wheels. Colossus read the cipher’s stream of characters
and counted the frequency of each character. Cryptographers then
compared the results to the frequency of letter distribution in the
German language and to a sample chi-wheel combination, continually
refining the chi-wheel settings until they found the optimal one.

The setting that gave the highest count of •s was found to be optimal (F. Carter, Codebreaking with the Colossus Computer, 2008, pp. 16-17).
They understood the settings of the wheels because of the extraordinary work of William Thomas Tutte, OC FRS FRSC. From Wikipedia:

At Bletchley Park, mark impulses were signified by x and space
impulses by •. For example, the letter &quot;H&quot; would be coded as
••x•x. Tutte's derivation of the chi and psi components was made
possible by the fact that dots were more likely than not to be
followed by dots, and crosses more likely than not to be followed by
crosses. This was a product of a weakness in the German key setting,
which they later eliminated. Once Tutte had made this breakthrough,
the rest of the Research Section joined in to study the other
impulses, and it was established that the five chi wheels all advanced
with each new character and that the five psi wheels all moved
together under the control of two mu or &quot;motor&quot; wheels. Over the
following two months, Tutte and other members of the Research Section
worked out the complete logical structure of the machine, with its set
of wheels bearing cams that could either be in a position (raised)
that added x to the stream of key characters, or in the alternative
position that added in •.