YD explained pretty early on in the movie that she was dismantling her military.
She had hardware to conquer/colonize/terraform *dozens* of planets. You can't get rid of all that shit in two years unless you just huck it into a sun; dismantling/repurposing it all would take decades or more.
It's rather plausible Spinel warped out of the garden, found a storage bunker on Homeworld with a bunch of gnarley shit including that big bottle of planet killer and an injector. The rejuvenator makes sense too since she was from a very bygone era and would recognize it/know what it does (Bismuth was the only other person aware of what it did and...I wanna say she's one of the older gems?)
...Or if you need a better explanation: Pink Diamond was just ready to receive her own colony and Spinel was with her up until she was abandoned. PD probably got a master class on colonization tools like the Injector and even the rejuvenator if her subordinates got out of line and therefore Spinel would have knowledge of them and she was clearly hurt and desperate enough to bring in the big guns the moment she got to earth.
Which was pretty fast after Steven's message. The timing is a ...little wonky, but Spinel certainly had the element of surprise going in so I don't think it's TOO much of a stretch especially if nobody was really guarding any of this shit (because of the dismantling efforts/lack of any known enemies.)