Do I get the hold effect from Grove after swapping the Brush Battlefield back to it?

No, you do not get the hold effect from Grove of the God-Willow immediately upon swapping the Brush battlefield back to it.

According to the rules for "Hold" (Rule 383.4.d), a Hold Effect is triggered by a unit maintaining control of a battlefield during the Beginning Phase when you score Victory Points from holding.

Here is the breakdown of why this does not trigger when you swap:

1. Timing of the Swap: Per the Brush battlefield token text, you swap it back when you "score here" (which is when you conquer or hold). The act of swapping the battlefield occurs as a result of that scoring action.
2. Timing of the Hold Effect: The Grove of the God-Willow hold effect ("When you hold here...") triggers because you met the condition of holding at the beginning of your turn.
3. Interaction: Since the swap happens after the scoring action (as a way to finalize the score/replace the battlefield), the Grove of the God-Willow is not the battlefield you were holding at the start of your turn. You were holding the Brush token, which then triggered the swap to the Grove. Because the Grove was not present at the beginning of the phase when the Hold action occurred, its "When you hold here" ability does not trigger for that turn.

In short, you must start your turn holding the Grove of the God-Willow to trigger its hold effect. Replacing a battlefield with the Grove after you have already scored at that location does not retroactively trigger the Grove's hold ability for that turn.