iMAC (Monterey 12.7.6/Intel i7) slow to boot up.

If it’s the 8 GB / HDD conflagration, err, configuration, it can be made appreciably faster for the cost of an external SSD, and shuffling some bits around.

Use an external SSD as your startup disk … - Apple Community

Three or four minutes seems long now that you have become accustomed to using your brand new M4 Mini.

It may be a minute or so slow but who cares. There is never a need to routinely shut down a Mac. They work best if you just let them sleep. Then, boot times don't matter.

If you're not going to use your Mac for weeks or months, go ahead and shut it down. Keep them connected to power though, because disconnecting a desktop Mac from power will prematurely deplete their NVRAM battery which gets expensive.

Now, one of the two (old ex-windows) screens on the new system has given up the ghost so I am looking at using the older iMac to replace it using a non-Apple solution as this is not possible using the Apple screen sharing procedure.

I'm not quite sure what you mean by that, but it has no bearing on slow startup times. If you remain curious I recommend using EtreCheck and posting its report in a reply here. Instructions: How to use the Add Text Feature When Posting Large Amounts of Text, i.e. an Etrecheck Report - Apple Community. It often provides some actionable information.

Consider booting your old Mac in "Safe Mode" Start up your Mac in safe mode - Apple Support, which performs certain actions that may be affecting startup times. Let the Mac take as much time as it needs to boot Safe Mode, log in as usual, use it for a little while, and then restart your Mac normally.

Usual reason for Intel iMac performance discussions is the 8 GB / HDD configuration:

Why is my hard disk drive iMac so slow? - Apple Community

Your old Mac probably has a slow mechanical (5400 RPM) drive and the new Mini is a fast SSD type which is MUCH faster than the old iMac.