Here are some questions anyone would need when attempting to troubleshoot anything tech related in general.
Basics:
What version is the end user's iPhone currently running. e.g. 16.4
What phone is the end user using with the issue. "iPhone 12 Max Pro". (Thanks for providing that)
This is not limited to just my phone. I see it's happening across iMessage
If this is so, What other devices is the end user accessing iMessage on. e.g. iPad running iOS 15.5 or MacBook Pro running 13.3. If this is not known, this can be seen from settings>general, about i<device>. or on a Mac book, Apple Logo> About This Mac.
More specific to the OP:
Speed tests:
Since the end user is having an issue about a potential lag when receiving or sending messages, it would be recommended to see what the current network status are on the devices.
For an iPhone please open up Safari and type "speed test" into google.com. On the search results there will be a widget that allows a user to run a speed test. (don't leave google) While only connected to cellular, Perform the speed test and see what the results are. Once that is performed then perform the same test by turning off cellular data off and connecting to wifi.
Example results:
"ATT" Cell at 5G or when testing switched from 4G to 5G etc. 90 mpbs.
Wifi network, 10mbps.
If the end user has access to a MacBook, hold the Option button on the keyboard down and "left" click on the Wifi Airplay logo at the same time.
Grab the following information.
Channel: number.
RSSI: - number dbm
Noise: number db
TXrate: numberbymbps
PHYmode: 802.llg (wifi standard being used.)
MCS Index: number
NSS: number.
speedtest on google search was 210mbps.
The end user would also want to go to google and perform the same speedtest on the google search.
Now the most important thing would be to have videos of screen recordings of the issue occurring ( Pictures and videos are great to see issues in action. ), or an explanation of how bad this "lag" the end user is experiencing. "E.g. When I send a message while connected to my Phone's Data, I get the blue progress bar and it takes forever or about 8 minutes for the message to send." or. " E.g. When I receive a message from another iPhone, the MacBook the end user owns get's the iMessage about 5 minutes before the end users iphone receives the message."
One more thing. When the issue occurs with the messages lagging, does streaming music, or playing a YouTube video on the phone buffer or have issues playing while the sending of the messages occur. and/or is the end user using a VPN service, try to toggle off the VPN and see if anything changes.
It is frustrating when an end user is performing basic troubleshooting and is still having issues, especially if after an update was pushed through and a feature or product stopped operating as designed.
I believe the people at Apple would want to attempt to try to find the root cause of the issue but from their perspective they have to look at every case as an new one and need all the data from the beginning; even if the issue was told before which could also be annoying. Nice thing is the beginning is a great place to start.
From My perspective of this issue, I am thinking there might be a network issue, or possibly and issue with the device itself, or not that I would believe so until more information was given, possibly the account? Although I or anyone else can't make a decision from what has been posted until some of this data is gathered to eliminate environment factors.
I hope that helps, with the information gathered it could help with finding the issue and receive greater attention from the developers at apple.
(Note: I am not an Apple rep.)