It seems to me there would be some level of Radio Emissions due to solar activity at all times. It might be possible to build or find a directional antenna
and use existing available radio equipment (microwave or possibly old AM-FM tuners) to locate the sun behind the thick clouds after the pole shift. If
we can locate the sun fairly accurately say within one to several degrees at different times of the day then we can determine many things. Where the
new rotational axis of earth is. What latitude we are at. From latitude we can determine estimated seasonal temperature range. Over time 6 months to
a year we can determine the new precession value or what the 23.4 degrees has now become. We can determine when the seasons (summer and
Possibly short wave moon bounce equipment would work in reception mode. One would find the noisiest spot in the sky and measure it's angle several times/day for several days. If this followed a sun rise sun set pattern over several days, then one would track it for 6 months to a year, say once a week or once every several weeks. From a pre-built excel spread sheet one would calculate the above needed information from the angles and time of day found.
The following is some quotes on some preliminary research on this subject.
1942 : J.S. Hey Detects Solar Radio Waves
One of the pleasures of radio science is its ability to reveal just what modest additions to the human sensory apparatus are needed to extend perception into entirely new realms of experience. As discussed in part 1, serious research can be conducted with a length of wire, a crystal earpiece or headphones, a tent pole, a hammer, a patch of earth, your ears, and a notebook. Similarly, given enough luck or patience, it's possible to detect some form of radio emission from the sun with virtually any radio receiver.
In 1942 British Army physicist James (Stanley) Hey was asked to investigate huge rushing noises thought to be hostile jamming of British radar stations. Hey sourced these mysterious signals not from the German Army, but from the sun, observing a strong correlation between radio emission and sunspot activity. In March my first attempt to renew the search, prompted by the highly scientific observation that it was a nice sunny day, was immediately rewarded by a huge swathe of solar noise roaring into the Lafayette at 13.8MHz: accompanied by the parallel 2nd harmonic - characteristic of a Type II emission   - at 7.29MHz.
Offered by Mike.