The door we want to use is a solid wood door at least 45mm thick, and was the front door of one of these massive old victorian buildings. It's really excellent quality.
Maybe so, but it ISN'T a fire rated door, it CAN'T be tested, so it therefore SHOULDN'T be used in a multiple occupancy building regardless of what you paint it with. The regs relating to fire doors are actually quite tight and apply to ALL new installations (and by definition this is a new installation). The rules about what modifications can or should be made to fire doors are meant to apply to existing doors in listed and/or historic buildings, so tradesmen (such as joiner) are no longer allowed to upgrade doors themselves and declare the work as meeting the regs. Will it make a difference? Does it really matter? You'll only ever find out if there is a fire, by which time it is way too late.
One question that you need to answer is "Who manages the building?" If the flat is subject to a lease the leassor (or management company) must be informed and they have a legal obligation to ensure that the work done is both safe and meets CURRENT legislation. If not, is there any legal contract or agreement in place between the two owners? Even if there isnt I think that you would be under a legal obligation to ensure that your works meet current fire standards.
I feel that you will also need to inform your insurers of these works and seek their approval.
TBH second hand door which cannot be fire rated is just a risky proposition regardless of how solid it is. Personally I'd just install a current spec FD60 fire door with appropriate hardware/fittings, hung in a replacement casing and have done with it, and then sleep better at night. In any case many enclosed shared access spaces in modern buildings require 1 hour fire doors which are generally 51mm thick as opposed to the 44mm of many 30 minute doors
For a previous employer I did act as one of the company's certified fire door inspectors, albeit on mainly commercial properties (offices, hotels, libraries, etc) and we were never allowed by Building Control to do the sort of thing you have proposed.