I don't have Hforman's long years in IT, but I worked with a major American corporation for twenty-five years and saw quite a bit of user difficulties, some of them their fault, more often the combination of a faulty upgrade and a user still trying to get the machine to work the way he/she remembered it working. What I learned from this is that complex problems, especially with booting, require more than one thing to go wrong at the same time.
You didn't give us the history of the machine, but the combination of Blue, Black and especially Red system failure screens suggests that at some time this computer had an entirely different OS on it -- perhaps Vista, perhaps XP, perhaps even a Linux distribution running on dual boot. If this configuration was not completely removed before Windows 10 was installed -- possibly as an update to the existing Windows boot OS -- this could cause some of the difficulties you have seen, and likely others. The Windows 10 OS could be attempting to load boot files from multiple but incompatible locations, resulting in the bizarre behavior you have seen.
If the data and, more importantly, the applications in use must be preserved, the indicated response would be to back out of every update in reverse order, right down to the de-installation of Windows 10 to restore what's left of the original boot sequence. I freely admit this may not be possible at this point; if your applications cannot be restored from installation disks or downloaded from the manufacturer, you may lose these apps and their data as well. But this process may reveal what has been hidden heretofore -- the prior boot order of the original setup. From there, you can determine what needs to be removed in order to leave a clean version of the prior installation of Windows to upgrade. This may require the deletion of partitions and their contents; you should be prepared to copy those contents to other media before proceeding, if these partitions do not contain the Windows operating files.
If you don't mind the loss of applications, and the data is backed up on another medium or not on the C: drive, you could simply reconfigure your BIOS to eliminate any prior modifications, including dual boot capability, and reinstall Windows 10 as if on a new machine. For this you will either need a working or at least readable partition with a licensed copy of Windows on it, or install as if on bare metal, erasing the C: drive in its entirety. This will clear your problems not unlike a D-12 bulldozer clears a field. You can then proceed with application reinstallation, if possible, and restoration of data.
What you will end up with is a machine with a single boot partition and no provision for dual boot to anything else. If this meets your operational need, you are done. But either way you go, you will spend considerable time and effort and risk considerable data loss. From the description you have provided, I suspect your machine has a complicated history and was not maintained or updated properly, which will certainly produce considerable difficulty in simply removing the layers of prior changes to reach something close to the original configuration.
It would be far simpler to put in a new C: drive and install Windows 10 to it, then transfer data from the old drive via an HDD docking system. This of course would not transfer the working applications, which would have to be reinstalled, if possible. This raises other questions of the machine's intended function and its desired compatibility with other machines in its network and/or its working group which I cannot address.
But I would proceed with your machine with the understanding that it may have multiple problems with its boot capability, as well as other software problems resulting from the original misconfiguration, and document carefully everything that was done to try and return it to an unconfused state -- if only relatively speaking.
Was this machine originally assembled and its software loaded outside Britain? This could also be an issue.