We have been calling the star by looking at which we can make out the direction of the North Pole the North Star. This was the only method at the disposal of the sailors during the days when they did not have any compass with them. When the axis of the earth would point toward Denab, Denab would become the North Star. When it would point toward Vega, Vega would become the North Star. Likewise, when the axis would point toward Thuban, Thuban would become the North Star. When we look at the Polaris from the earth, the Big Bear appears to be revolving around it. So I am curious to know which constellations would appear to be revolving around the Denab, Vega and Thuban when the axis of the earth would be pointing toward them due to wobbling of the earth.
When we look at the Polaris from the earth, the Big Bear appears to be revolving around it.
Ursa Major is relatively close to Polaris in angular distance, so this motion is easier to see. But in fact all the constellations appear to revolve around the axis. The closer they are, the tighter the arc and the easier this motion is to see. If they are far away and you are not at very high latitudes, they will be obscured at some point by the horizon and the motion is hidden.
So for your question, look at a starmap and the closer a constellation is to the future axis, the more it will behave like Ursa Major today.