I believe you'd be best served to find someone that has one and try it.
Now, I have never shot an Uberti birdshead grip (that would be their Pinkerton revolver, I believe). I have, however, shot these.
The Classic Arms Ace.

When I shoot it it rotates in my hand until it is pointing straight up in the air. Since it's a single shot, that don't matter. The lack of a trigger guard may help the rotation, as does the smallness of the grip.
The Pedersoli Kentucky Pistol.

It does not rotate up like the Ace does. Larger grip? Trigger guard? Longer, heavier barrel? I don't know what keeps it down, but it doesn't rotate.
Then there's the Uberti Cimarron Thunderer.

There's a nice bulge in the middle of the grip - it's fat, easy to hang on to. And the shoulder at the top helps prevent rollup.
But the Pinkerton, while it appears to have the bulge, doesn't have the shoulder.

This has made me leery of rollup, so I've never tried one.