Yes, the stomp attack is available after the trip
The text says simply "if the target is prone" (MM p. 322); it's not conditional upon the target being knocked prone by the gore attack at the end of trampling charge. All that the battlemaster-druid needs to do is hit with a weapon attack (PHB p. 74) which the gore attack is.
Gore: melee weapon attack. (MM. p. 322)
Let's test this against another case: your ally knocked the target prone.
Your ally uses an action to shove/knock prone.
Your elephant attacks with gore. (and / or) ...
Your elephant attacks with stomp.
Stomp: Melee Weapon Attack. +8 to hit, 5' reach; one prone creature.
Hit 21(3d10+5) bludgeoning damage.
Both attacks work since the target creature is present and prone. The druid didn't need to use the gore attack to unlock the stomp; "something else" knocked the target prone.
Does timing of the saving throw(s) matter? No.
The targeted creature is dealing with two different problems: (1) the trampling charge+gore attack and (2) that "thing battlemasters do" which trips on a hit. The rules don't constrain the battlemaster in this case. The battlemaster gets to choose, and makes this choice contingent upon a hit. A key subclass ability here is the player's choice to expend a superiority die, or not to.
Regarding simultaneous effects, this is addressed in Xanathar's Guide to Everything (p. 77):
If two or more things happen at the same time on a character or monster’s turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character’s turn, the player decides which of the two effects happens first.
To get down in the weeds a little bit, inside the battlemaster-druid's head is a decision tree made at combat speed; I hit 'em; still standing? OK, do "this cool thing I do" and to knock 'em prone! The save or failed save from the trampling charge gore attack would be evident at impact if we are going for versimilitude here.
Besides that, combat maneuvers are what makes the Battlemaster cool and unique as a subclass; using them comes at a cost (see below).
The six second combat round has "a lot of things going on at once." The elephant-druid-battlemaster's turn isn't over if someone else failed a save. The trip attack isn't conditional on anything but a hit.
"... you can expend one superiority die to attempt to knock the target down." (PHB 74).
No timing is specified other than "hit" the target; what the target is doing (saving or not) isn't a constraint.
- What does the first save represent? Ideally, it happens as the trampling attack+gore hit occurs; the "did it or didn't it get knocked prone" result happens as close to the gore as can be imagined. The trip attack is "something else" that happens during the course of the druid-battlemaster's turn that (potentially) knocks the opponent down.
An argument against
The trip attack isn't a bonus action, but is triggered "when the attack happens;" a DM could rule that the player must wait for the next round since the results of the trampling charge+gore attack, and a save against it, have already taken place.
Yes, this is a nice nova. The example case isn't available until 11th level. (Moon Druid 8 wild shaping into a CR 4 creature + 3 levels of BM).
A price was paid for this!
The player in the example case has given up on one 4th, two 5th and one 6th level druid spells, to get four opportunities per short rest to do "a cool thing!" which can fail, since monsters can make saves as well as miss them. The druid has also given up, for the time being, Elemental Wild Shape which arrives at 10th level druid.
Druid spell slot progression
