Speaking from personal experience, I'd use a word that doesn't have different associations in several parts of the world.
Some time ago, I had to have a high school diploma translated, so I know how hard it can be to find the right words. I believe a lot depends on the customer's input (e.g., you need to have the school's description for many classes in the US because their titles don't tell the whole story).
Take writeaways's second link:
https://www.treehoppersfk.comIt only uses Kindergarten as part of the
proper name of the school. Elsewhere, they call it woodland nursery. While you don't have much space, maybe you could write the name of the preschool in German and add a word or two in parentheses.
Forest School isn't that wrong either. Have a look at the chart and first para. here:
https://www.forestschoolassociation.org/history-of-forest-sc...This has little to do with the German Waldschule on Wiki. It even says Denmark in there. Not sure why some treat Wiki as gospel.
The BBC calls them forest nurseries:
https://www.bbc.com/news/uk-england-london-42059794I can see why you asked that Q, as there may be more than one answer. But keep it simple, IMO.
Best