It seems that nobody on the internet is certain how to pronounce it. This video is the only thing I could find. It says Cosmoem is pronounced coz-MO-em. This pronunciation makes sense if we look at the (possible) origins of Cosmoem's name, as well as it's German and French name.
Cosmoem may be a combination of cosmo and embryo.
This makes some sense. An embryo is an organism going through development until it emerges into the real world. Something that immediately comes to mind is my childhood years of The Crocodile Hunter and Zaboomafoo, where they would discuss reptile/bird eggs and how the embryos developed inside them. This also makes sense in relation to Cosmoem, who appears to be an egg, inside of which Cosmog starts developing into Solgaleo or Lunala.
Cosmovum may be a combination of cosmo and ovum (Latin for egg).
The German and French name for Cosmoem further supports the above deductions with its ties to eggs. I'm sure, at this point, you've noticed that Cosmoem is thematically more like a cocoon than an egg, but Game Freak isn't 100% logical so I still think the embryo/egg explanation makes sense.
In conclusion, Cosmoem is likely pronounced coz-MO-em due to it being a combination of cos-mo and em-bryo. However, in the grand scheme of things, how you pronounce Cosmoem is up to you. Similar to how I pronounce Scizor as SYE-zore while other people pronounce it like scissor.