A "magic weapon" includes any weapon that has an enhancement of any sort beyond the mundane - so the everburning blade would certainly count as one. Even a Weapon of Warning - which has no bonus to hit or damage at all, is still a magic weapon. It's not uncommon for extremely low level parties (1-5) to sometimes come across magic weapons that have a minor enchantment (often a cosmetic effect), which are functionally +0 magic weapons - no combat bonus, but magic for the purposes of getting through resistance.
Weapons that have potency by way of their material are an edge case, and different DMs will rule differently; if the material is magical in some way, or not natural or not native to the material plane, then the weapon will usually be considered magic as well, by virtue of being made out of that material - but in cases where your world space has powerful materials available in the mundane world, not classifying them as such is more understandable. In the Realms, Mithral and Adamantine are considered magical materials - items made from them are considered magic items, by virtue of being made from those materials.
Pure raw, one person could cast magic weapon on a mundane sword - after that, it would not be an applicable target for a second person to cast the spell on, because it would not be a nonmagical weapon at that time.
If we were to remove the non-magical weapon restriction, then the rules regarding effect stacking come into play - when multiple instances of the same effect are applied to a single target, they do not stack, and the strongest version is used - so if you cast an upcast magic weapon on a sword that had a base level magic weapon applied to it, the new spell would take effect, and the bonus would increase to the bonus provided by the stronger spell. If the weaker spell somehow had a longer duration (sorcerer extend spell, for example), then it would still be present on the item, just not in effect, and when the stronger spell wore off, the weaker spell would reassert itself and continue for the remainder of its duration.
- Paladin with Meta-magic adept casts an extended Magic weapon on her sword; it gains a +1 bonus and she can concentrate on this for 2 hours.
- Half an hour later, her ally casts a 6th level magic weapon on the blade; the sword now has a +3 bonus, from the 6th level spell, and the 2nd level spell is suppressed.
- An hour later, the 6th level spell runs out; the sword drops back to having a +1 bonus, from the first spell - the Paladin is still concentrating on this, and has half an hour left still; her spell continued to count down while it was suppressed, but it has not expired yet.
