If you bump up the randomTickSpeed to a high number, it does not speed up the process of cauldrons being filled by rain.
Expected:
Rain is on, a grid of cauldrons been placed, not snowing, not a desert.
randomTickSpeed = 3: slow filling.
randomTickSpeed = 3000: fast filling
What happened:
randomTickSpeed does not affect it at all
it would make sense cause it randomly fills the cauldrons the way crops grow.
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Affects 1.16 Release Candidate 1
Affects 1.16.2 Pre-release 1
Can confirm in 21w03a.
Can confirm in 21w05b.
Can confirm in 21w06a. Video attached.
Can confirm in 21w07a.
Can confirm in 1.16.5 and 21w08b.
Can confirm in 1.17.

Can confirm in 1.17.1 Pre-release 1.

Also affects powder snow.
Can confirm this behavior in 21w40a. Here are some extra details regarding this problem.
The Bug:
The process of precipitation filling cauldrons is not affected by the "randomTickSpeed" gamerule.
Steps to Reproduce:
Enter a biome where it rains or snows and summon lots of cauldrons.
/fill ~1 ~ ~1 ~15 ~ ~15 minecraft:cauldron
Set the weather to "rain", and take note of how fast the cauldrons are filled with precipitation.
After waiting for a fair amount of time, increase the value of the "randomTickSpeed" gamerule in an attempt to speed up this process.
/gamerule randomTickSpeed 300
Take note of the speed at which the cauldrons are filling up now, compared to how fast they were filling up before.
Observed Behavior:
The process of precipitation filling cauldrons is not affected by the "randomTickSpeed" gamerule. No matter what the value of the "randomTickSpeed" gamerule is set to, cauldrons will not fill up any faster with precipitation.
Expected Behavior:
The process of precipitation filling cauldrons would be affected by the "randomTickSpeed" gamerule.
Can confirm in 1.18.
Can confirm in 1.18.1.
Can confirm in 1.18.2.
Affects 22w18a
Can confirm in 1.19.2.
Can Confirm 1.19.3 Release Preview 3, problematic for gameplay on custom maps/servers.

Can confirm in 23w03a

Can confirm fixed in 23w31a
This issue was present in 1.20.1, but no longer occurs in 23w31a. This issue has been fixed in 23w31a so I'll be resolving this ticket as fixed. To support my claims here, I've attached two screenshots.
[media][media]
Can confirm for MC 1.12.1.