Drying time will depend on the diameter and length of the birch log.
Birch bark can be thick, and will really slow the drying time. If left too long in a warm, humid atmosphere, it will start to spalt quickly. You can prevent this somewhat by cutting shorter logs or poking some holes into the bark. Be sure to leave air flow between the logs. Use little sticks for spacers. Some people like spalt, some don’t. I would assume that since you are going to put them on a wall, you probably don’t want the spalt, and would prefer clear, unblemished wood.
As the wood dries, it loses moisture, shrinks, and separates from the bark because the bark does not shrink, no matter what kind of wood it is. Depending upon the type of wood, how much sap and moisture was in the wood when it was cut, and what time of year the tree was cut, the log may check and crack while drying. Each log is different.
When I want to have a perfect birch log slice, say about 6” diameter, I will only let it “dry” for a few weeks, if that. It works out best for me to cut them green (wet), and then soak them in Pentacryl. Pentacryl is a non-toxic solution produced by Preservation Solutions that replaces the water in the wood, and helps keep the wood cells from shrinking. I have been using it for years, and it works particularly well on birch. It may slightly discolor the bark, but not really noticeable. It also has a slight odor, but that dissipates quickly after dried. They also have a “Wood Juice” product made for dry wood. Go to their website and read more about it. preservation-solutions.com/wood-treatment-products
After the slices have been thoroughly coated or soaked with Pentacryl, then allow the wood to dry. Drying time will depend on the thickness of the slice. Keep the slices separated. Use an old grate or oven rack or something to stand them up so they don’t touch while drying. You can use a fan to speed it up. After the Pentacryl has dried, you can coat them with a varnish or anything you like.
If you don’t want to use Pentacryl, and coat them with a varnish instead, any oil based varnish would work, but coat both sides of the slice. If you only coat one side, humidity is drawn in from the other side if not sealed and may warp the slice if it is thin. Indoor applications can be coated with a water-based varnish, but the wood must be dry or the moisture in the wood might turn the coating white, or cloudy.
Or just go the easy route. Let the slices do what they do, crack, or whatever, like a pile of firewood, glue to your wall, tack or nail if necessary, and varnish the whole wall later. This looks great on walls next to an electric fireplace to make it look more real.