The calculation of a fix from GNSS is actually rather complex and the potential complexity is almost unlimited. The most accurate GNSS fixes are obtained from offline solvers that analyze long-duration (e.g. 24 hour) recordings of the GNSS data streams and use computer models to determine the most likely solution accounting for many types of error. This is all to say that there isn't really one approach to calculating a fix from GNSS, and receivers embedded in devices like phones use pretty simplified methods so that they can run in real-time and with a short time to first fix.
Most GNSS are largely similar, though, and there are certainly enough papers describing how to perform fixing across satellites of independent constellations so I'm sure it's been implemented in at least higher-end devices. Still, I suspect inexpensive consumer receivers are probably calculating a fix per constellation and then combining those. Fixing across constellations requires combining ephemera, time base, etc. data that varies between them, but all of these are known with high precision, you just have to pull them together.
Since most GNSS are mostly the same, new GNSS constellations are usually launched either to improve coverage in a region, or these days with several global constellations, more often for sovereignty reasons. If you have weapons systems that rely on GNSS, it's better to have your own constellation. GPS for example no longer has a wartime selective availability capability (that would deny use by anyone other than US forces) but it did in the past and, if WW3 broke out, it would probably be implemented again. For much this reason different GNSS constellations tend to operate in different bands (not always, though) so that a country can selectively jam adversarial GNSS systems without affecting their own. GNSS has its origins in the military and is still largely a military capability today, with Galileo being the odd one out that focuses more on commercial applications (Galileo's selective availability capability is used to charge a subscription for higher-precision uses).
And some GNSS do have "accessory" capabilities, probably the most prominent being Galileo's two-way search and rescue messaging capability that will compliment COSPAS-SARSAT. But that's always been the way with satellites. For example, the GPS constellation carries detonation sensing equipment for nuclear counter-proliferation. It just saves money to combine payloads.
One of the reasons to not put GNSS constellations in separate bands is that it simplifies the construction of multi-constellation receivers. Of course, it also makes it more difficult to selectively jam. Some combination of these interests mean that GPS, GLONASS, and Galileo are very close siblings in the radio spectrum. It gets surprisingly political.