Well I found this:
Basically every Pokemon on a route can call 7 "SOS" Pokemon (Slots 1-7). The exact rates aren't known yet, but the Slot 7 Pokemon is by far the most frequently called, I suspect around 2/3 of the time. It can also randomly call any of the Pokemon in Slots 1-6. It might be that Slot 1 is the rarest and they become increasingly common, or Slots 1-6 might all be equally likely. Don't think we know yet.
Source, Here is the table for Kala'e Bay on Seribii.
I will try to explain this, but I am not the best at explaining things. :P
First, make sure you are SOSing on Ultra Sun/Ultra Moon in Kala'e Bay, because that is the only place that Slowking can be SOSed (and not in Sum/Moon).
Second, there are 7 SOS slots, which SOSing Pokemon Pick the "initial 7th sot" first turn and then randomly chooses a slot (except the initial) whenever they SOS a Pokemon again. Pokemon in SOS slot one are usually the same Pokemon SOSing, while Pokemon in slot 6 are Pokemon Like Salamence and Slowking. This makes sense because SOS slot 1 is the most commonly picked, whilst the other ones are more rare.
Now math time. Slowking has a 20% chance of being SOSed from a Slowpoke on the rarest SOS slot (slot 6), and it can only be found in that slot, meaning that 20% chance is really much lower, depending on what slot is picked. Lets assume the quote above is correct (because rates are not yet known) in the First Slot being 2/3 of the time. From here, lets guess that slots 2-6 have an equal chance of being chosen. With that estimate, the chance of slot 6 (Slowking's slot) is
1/3 divided by 5, which is
1/15. Then, we must find
20% of 1/15, which can be done as such:
1/15 divided by 100 Times 20 (turn 1/15 into 1% of it and then multiply it by 20 to get 20% of it) which comes to
1/75 or a 1.333333333% Chance of getting a Slowking.
TL;DR: We don't have the exact numbers, but a rough estimate (1/75; without factoring in whether slowpoke SOS's or not) suggests it is quite difficult to find, but way more likely than getting a shiny. This also checks out with many people saying it took them well over 200 SOS's to find a Slowking. Sorry about the odds, it is hard. :P
Hope I Helped!
Edit, fixed a mistake in misinterpreting the "Initial Slot" :P