Advice request : simultaneous function/data updates on many databases

We have many databases of the same type separated for data governance
reasons. They, however, share the same web front-end code.

Presently, replacing functions and performing data updates on the
databases in series often executes across all databases in less than a
minute. (The updates are currently done with simple sql files connecting
to each database and then loading a stub file pointing to each function
to drop and reload, and running the data update queries.)

However, for larger updates, the time when the front end code is
out-of-step with the database can cause end-user problems.

Unfortunately our schema arrangement isn't clean enough to swap out
function schemas in a transaction to sort out that part of the problem
(if in fact that would work anyway).

One solution might be to do the updates in parallel. Another thought
would be to somehow execute the code update from a text field in a table
in each database triggered with pg_cron.

Bearing in mind the possible problems of connection saturation or
massive IO spikes, I'd be grateful to learn of any thoughts on how to
negotiate this problem.

Rory