I would reveal the wood and consider a runner.

I would vote wood of course but you may be surprised how pricey it is to use wood rather than carpet.

absolutely agree to remove the carpet from the stairs if it will not continue on either floor. If cost is of concern, you could consider continuing the tile for the stairs. Or you could only lay wood on the stair tread and leave the stair risers open.

yes it will look lovely with wood on the stair .

The wood would look beautiful. However, it can be quite slippery. Something to take into consideration for children or pets. And it can be quite costly. The cost is in the labour, not the material. If you are looking to tie in the wood, keep the stairs safe, and keep a budget - a good recommendation is to do a "fake runner". Essentially have approximately 6" of wood on either side of the stair, and run carpet up the centre.

I would actually do laminate if I were you but they can't get very wet

Wood is the way to go! Unless you want to sleep on the stairs : )

If money is no object go with wood, but stairs are built very sturdy, to prevent noise and flex. The carpet blends with the tile, so if you need a few grand someplace else leave it. Being a woodworker I would check how it was made to see if a saw all could be used or if glue and wedges were used, making rebuilding complete stair a must.