I have been checking my house circuits and came across a run that is made up of 14/3 on 2 separate fuses, which branch off in the attic to two runs through the house. I don't like the idea of the neutral being shared throughout. I drew a picture of what I found and am wondering if the solution I have come up with looks ok to some of you professionals. And Hello from Nova Scotia. I thank you for any input in advance.
As long as the circuits are on different phases there's a decent chance this is safe as is, but it's a bad idea to wire this way in a home and a good idea to fix it. You can't be sure of what's behind the walls but your solution might completely repair the situation and will be no worse. Better to attach your new run to an outlet than a light. If you use a light be aware that some lighting circuits are powered from the ceiling box, others are powered from the switch box. Make sure to connect to an unswitched point in the circuit.
The style of wiring you found is common in large commercial installations. It is simpler and cheaper to reduce the number of neutrals running around the place by 1/2 or 2/3. If using conduit and individual conductors it saves pulling extra neutrals for nothing. But the key is that it has to be designed and maintained properly. All the circuits sharing a neutral have to be on different phases. If an installation is designed so that ALL its circuits are run this way, and if professional electricians do all the maintenance over the years and decades, this will work fine because everyone knows what's going on. But in a home, those are terrible assumptions. If one of the breakers on a circuit like this fails, and is replaced, and the new one is on the wrong phase (because the homeowner doesn't know it was done this way or even a pro doesn't realize) then the neutral will be overloaded and the breakers won't protect it. Also, if a homeowner turns off one circuit to do maintenance he won't realize that there are live wires in the junction box he's servicing or that there could be a dangerous voltage across a broken neutral during servicing.
What you're describing is a multi-wire branch circuit. It's totally fine - I don't agree with the fear of them. Even better, your neutral splits for good prior to the first receptacle or lamp. That means you don't have several worries that often come with MWBCs.
For instance, you do not need to pigtail neutrals because all your outlets (including lamps) are downstream of the split. If you removed the master B/R light and severed the neutral, that would not affect the bathroom et.al.'s neutral.
In the US we have a "belt and suspenders" requirement that both sides be on a single shut-off switch, which turns off both sides together for maintenance. I am being very specific: this is what the rule says. It says nothing about a ganged-trip circuit breaker (though that is a convenient way to do it if you use circuit breakers). Fuses are fine. It's not about overcurrent but worker protection. However any competent electrician working in the green box will recognize the MWBC split. I would not lose any sleep over a lack of ganged shut-off, since none of your outlets have both hots.
You absolutely do need to make sure the two "hots" are punched down to separate 120V legs, so the neutral carries only the difference in current. Measure the voltage between them; it must be 240V. If it is 0 volts, that would mean the neutral would carry the total of both currents in both circuits, which could overload it by 100%.
If I were you, I'd continue the MWBC in service unless you feel a burning need to install AFCI breakers (this being the only reason I don't use MWBCs). If you need GFCI, use a receptacle and put it downstream of the split somewhere. A plain GFCI receptacle works fine as long as it's downstream of the final split, which all your outlets are.
That type wiring is allowed in some locations but not where I live. It is okay if the circuits sharing the 14-3 are from each side of the input buss. In theory, the amperage draw on each circuit cancels the amperage draw on the shared neutral. 10 amps on the left circuit, 8 amps on the right circuit, equals 2 amps on the neutral. It saves the electrician a couple bucks but shows just who he is. He would not do wiring for me.