How do I restore seamless keyboard and mouse sharing between two MacBook Pros and an iPad after upgrading to macOS 15.5?

I have one 2019 Macbook Pro and one 2024 MacBook Pro. Both were upgraded to Sequoia 15.5 yesterday. It was only a point-release upgrade for both machines, as they were already on identical versions pre-upgrade.

Before the upgrade, I could share my keyboard and mouse between the two machines and have one MacBook working with an external display as well. So, I had three screens:

Machine A - the MacBook built-in display.

An External Monitor - attached to machine A and extending the display.

Machine B - Displaying its work on the Built-in display.

I could seamlessly move my pointer over to any of the displays and operate either Machine A's workload or Machine B's workload, depending on the location of the cursor. SWEET.

A similar behaviour could be observed with my iPad.

Now, it doesn't work. There is no way to get the cursor and keyboard to move seamlessly across to Mac B. If I mirror or extend Machine A, it will work, but I'm only dealing with Machine A's workload across all three devices. That wasn't the case before.

I have verified the following and am confident that the values I list here are accurate on both machines.

AirDrop and Handoff - "Allow Handoff between this Mac and your iCloud Devices" Toggle on. All devices are signed into iCloud (except the dump monitor, that is).

System Settings > Displays > Advanced

"Allow your pointer and keyboard to move between any nearby Mac or iPad" toggled on - both machines. This is what fails.

"Push through the edge of a display to connect a nearby Mac or iPad" toggled on - both machines.

Wifi - both machines are connected to the same network.

Bluetooth - turned on both machines.

This all worked great, and it was magic. Before the upgrade, Machine B wasn't displaying on the monitor, but, as I mentioned, it displayed on its Built-in display, and I could control it to perform the necessary tasks on that machine.

I have tried all sorts of configuration changes to the above and also included trialling the mirror (or extend) options but can't get back to living the pre-upgrade dream.

[Re-Titled by Moderator]