What is the best way to apply felt to wood?
Spray adhesive is a good match for this application. If you used the lower strength type, you might even have a decent chance of getting the felt back off later if you needed to for some reason.

Using spray allows you to get an even coat over the entire felt surface, avoiding lumps and/or hard-spots in the finished surface. The hard spots can arise where a glob or line of liquid glue is absorbed by the felt.

Spray adhesive generally has two modes of use, join surfaces while adhesive is wet, or apply to both sides and join surfaces after a drying period. I would go with the former in this case as the wet glue would allow you just a short period to work out any wrinkles. Also, it may reduce the holding strength, which will be much greater than needed anyway. That way if you ever need to remove the felt, you might be able to do it without shredding it.

Scotch-brand 77 is a common choice (pretty high strength though), and even mentions felt on the label: https://www.amazon.com/dp/B00114RCP0/