It gets to keep engulfing.
Here's the relevant part of the Gelatinous Cube's Engulf action:
Engulf. The cube moves up to its speed. While doing so, it can enter Large or smaller creatures' spaces. Whenever the cube enters a creature's space, the creature must make a DC 12 Dexterity saving throw.
On a successful save, the creature can choose to be pushed 5 feet back or to the side of the cube. A creature that chooses not to be pushed suffers the consequences of a failed saving throw.
So what doesn't happen is that it uses its action to try to engulf you, and then can't do it again this turn.
What happens is that it gets to move the entire 15 feet, and attempts to engulf anyone in its path. If it starts its turn within 10 feet of you, and there's no space to dodge out of its path, it can force you to make multiple Dex saves to keep running away.
This is its hunting strategy. The Cube is an ambush predator. It can't catch victims who have any space to run; it needs you to get as close as possible before it moves. The closer you are, the more Dex saves you have to make to escape. It has a few tools to support that strategy:
- Transparency makes it's hard to see when it's not moving.
- Blindsight lets it lurk in poorly lit places where it's even harder to see.
- Its shape lets it block a narrow passage to prevent prey from escaping.
The best case for the Cube is that some sucker blunders right into it. The victim will have to make one save with disadvantage immediately, and then be surprised, which gives the Cube another three tries to engulf them on its turn. So they have to pass five Dex saves back to back. This isn't even an unlikely scenario if it can hide in a dark spot that's accessible to surface-dwellers. If it's in total darkness, it may well be impossible to detect.