Mystra's Weave and the Multiverse
Using the Planescape parlance, this question is in the style of a "clueless prime". The Planescape setting is actually an over-setting, covering all of the other settings: even those that are isolationists like the Dark Sun or Ravenloft. Forgotten Realms is just one prime world amongst many others. It would be meaningless to assume that Mystra is responsible for the functioning of magic in the whole of the multiverse, let alone on, say Krynn or Mount Olympus or Oerth.
Each prime material world has its own space around it, called a crystal sphere. Toril is in Realmsspace, Krynn is in Krynnspace etc. Outside of the crystal spaces is some matter called phlogiston. A spelljamming ship can travel in a given crystal space (which allows people on Faerun to travel to Toril's moon Selune, for instance) or between crystal spaces (so you can go from Toril to Oerth, etc.). All of the magic in the Realmsspace is taken care of by Mystra. But beyond it arcane magic works by the rules of the particular sphere you go. (And it works without divine intervention in the phlogiston, otherwise spelljamming would be toast.)
Finally, spelljamming cannot get you beyond the prime material plane. If you go to other planes like Bytopia or Ysgard, etc. by means of a planeshifting spell, all those planes have their own rules and mechanisms for magic as well.
In short, no, Mystra and Toril's weave are not needed to cast spells beyond Realmsspace.
Generic Weave of 5e
There was no mention of the Weave as the source of all magic in the D&D multiverse before the 5e. It was a FR construct perceived as the embodiment of the goddess Mystra. Even in 5e, Ed Greenwood, the creator of the FR setting has tweeted:
The Weave is Mystra, and extends beyond Toril only so far as individual beings who are Weavemasters do (example: the Chosen, and Mystra herself when she went into the Hells), and even they can't access the full Weave when they're outside Realmspace.
Yet, the 5e Player's Handbook now identifies the Weave as the source of magic everywhere. This generalisation might have been introduced in order to facilitate the return to Vancian spellcasting in 5e after the 4e. If you think about it, declaring the Weave as the source of magic everywhere is quite unnecessary. It explains something supernatural, magic, with some other supernatural thing, Weave.
Imagine we had no FR, removing the discussion of the Weave from the Player's Handbook would have almost zero impact on D&D. Just for FR, it was part of the whole lore around the goddess Mystra, and no such baggage needs to be carried to the other settings.