fish, version 3.3.1
Linux arch 5.13.7-arch1-1 #1 SMP PREEMPT Sat, 31 Jul 2021 13:18:52 +0000 x86_64 GNU/Linux

First, I do not expect anyone to work on this. It is a largely frivolous bug that effects very few users.
I've been sitting on it for a while, but decided I might as well put it here for informational purposes.

Clean fish, with the following defined in fish_prompt.fish:

# Defined in fish_prompt.fish
function _refresh_prompt --on-variable foo
    commandline --function repaint
end

function fish_prompt
    fish --command "set -U foo $PWD" &
    echo $foo
end

Here's what it looks like in Fish 3.2.2:

In Fish 3.3.1 there seems to be a few hundred millisecond lag:

Weirdly enough, this works in Fish 3.3.1, and is perfectly responsive:

Maybe this means the bug has something to do with pgroups?

Alright so there it is. Unfortunately, I wasn't able to bisect the bug because of a separate bug that happened in between, #8088