My first thought was like Greg's, 3M spray on contact adhesive. But you have to use it judiciously (big word for early in the morning) so you don't soak the felt through. If you over apply the glue to the felt, you'll get a hard, flat spot that doesn't look right compared to the rest of the felt. When you apply the felt, cut and test fit it before applying glue. Felt will stretch a little as you work it. Make a center line on your wood and when you apply the felt, place the fold on the center line. Then carefully lay the felt down, working it with a small wooden roller like you use for flattening wallpaper seams. Work from center toward the edge, then work the opposite side. Try not to stretch the felt too much but you might still have to use a razor blade to make final trims along the edges.
They also make a felt that already has adhesive applied to the back. Haven't used it but certainly might be easier than spraying 3M and probably less risk of stretching out of shape.