The mice must be assisted across a gap too wide for all but the most skilled to negotiate alone, but the Shaman's options to help the mice are limited by the Shaman being trapped in a cage high above.
You can either wait for the Shaman or Air Jump and Wall Jump your way to the cheese if you're experienced. An extremely experienced player may be able to Corner Jump their way back with the cheese, thus making this map one where the Shaman is not absolutely necessary to complete the level.
One can build a support structure under the Shaman compartment and construct a bridge on top of that, then launch the bridge into position by shooting a ghost right-side-pointing cannonball at the bridge. Use the arrows to get mice out of the way of the bridge before launching the bridge. This is known as the triangle.
It is also possible to use the arrows to group up all the mice on the edge of the main log. Use spirit to help the mice jump over the gap and get the cheese on the right. Use the arrows to once again group up all the mice for the jump back. Use spirit again and get the mice back over the gap towards the mouse hole. A well placed cannon shot can also launch mice over the gap, but spirit must be used on the way back. A cannon can go right (direct shot) or down, to propel the mice by sliding off their side. However, in both cases, it is rather likely that some mice will die.
Anvil god is not an option to help the shaman escape. Conjuration glitch, however, can.