A very typical thing for a trade sign anywhere in Europe was to have a representation of something, because not everyone could read. Ergo, "The Grapes", ( a bunch of grapes carved from wood, painted, hanging from an iron bracket), "The Jumping Deer", ( a deer outlined in thin wrought iron, hanging from an iron bracket) You can find some examples of iron ones in D'Allemagnes' Ornamental Ironwork, published by Dover, and some are posted here.

Iron would last longer than oak, but of course it would be far cheaper to have someone paint a picture of a deer than execute one in iron. Oak weathers reasonably well, but if you intend to paint it, you'll want to fill the grain first with a paste filler, or you might not get quite the fine definition you want in the brushwork.

It's a little later than Tudor, but there are a lot of commercial signs up at Colonial Williamsburg that are painted wood. (As a theatre carpenter I have built replicas) Probably a couple of planks joined together, not plywood like it would be today, but otherwise not very remarkable.

Speaking as a woodworker: don't get too hung up on the specifics of "English Oak" It would have been made from that material if that was readily available. If some other suitable kind of wood was available, that would have been used instead, especially since the whole thing was going to get painted. It is easy to romanticize the wood choices that were made in 'olden days', but a quick look at the back or under side of any antique will show you some brutal economic choices being made. This is technical knowledge, so I'm not sure how I would cite it, but I can find photos of antique furniture if that would help.