They cast Druid spells, not Magic-User Spells.
Really depends.
So, the first D&D Bard class was published in
The Strategic Review #6 (or Volume 2, Number 1; the issue has both numberings on its cover), February 1976. Page 11. This very first Bard class casts Magic-User spells. Oh, by the way, for this version -- "Elves, Dwarves,
and Hobbits may be Bards but cannot progress beyond the 8th level (Minstrel)". Which is to say, any of the original D&D races could become original D&D bards.
Now, when Gary Gygax put together the AD&D
Player's Handbook, published in 1978, he included every single class from both the various D&D Supplements and from
The Strategic Review. This involved some minor adjustments for most of the classes, but he
radically reworked the Bard into a class that involved progressing first as a fighter, a thief, and then as a "bard", the last casting druid spells, and limited the class to humans and half-elves.
But lots of people preferred the original Bard, which was republished in
The Best of The Dragon 1 in 1980 (a collection reprinted, with changed ads, in 1985). So, during the 1980s, there were two in-print versions of the Bard class available from TSR, one in the PHB that cast druid spells, and one in a more obscure publication that cast magic-user spells. While officially people were told not to use that second class with AD&D, only with D&D, plenty of people went ahead did (and after the release of B/X in 1981, it was a much more natural fit for AD&D rules than what D&D had become anyway).
And then, of course, in February 1989 the AD&D 2nd Edition
Player's Handbook came along, and
its Bard was very clearly based on the one from
The Strategic Review, complete with casting what were now Wizard spells, but the class retained Gygax's limitation to humans and half-elves.
So, halfling Olive Ruskettle was first published as a novel character in October 1988, with her first game stats being published in April 1989. Which means that, sure, when she first appeared in print, official AD&D Bards cast druid spells, but by the time she was statted up, official AD&D Bards cast wizard spells.