Civilization – Would a city get the bonuses from an adjacent natural wonder

Here are the bonuses from being a Suzerain as listed in the wiki:

• The city-state follows you into war and peace with other players.
• Your units may enter city-state lands.
• Your Builder units may improve city-state tiles.
• See 3 tiles around the city-state.
• Can pay Gold to temporarily take control of the city-state's military.
• Gain ownership of all the city-state's resources.

• Not listed but you are also give their unique bonus for being their Suzerain. All the city states and their unique bonuses are listed -> here.

You specifically asked about the last one, gaining ownership. This is the same as it was in prior Civ games. By being the suzerain you get the luxury and strategic resources that they have. The luxury ones provide amenities (if you don't already have that resource), and strategic ones can be used for the corresponding units.

In no Civ games including this one do you ever get food/production resources from being the Ally/Suzerain of a city state. They just list that you control all resources, but the excess bonus resources do not give you anything as they are for providing bonus production/food on specific tiles.

If you consider only the points you specified, the answer is simple. Just build the mine :).

There are only 2 reasons you should be considering while doing this (hills with forest - bear in mind that lumbermill can also be constructed on tiles without hills but with forest):

• Religion that gives bonuses on forest tiles (I'm speaking from memory but afaik there was one bonus like that)

• Buildings that require adjected forest (I think there was a wonder that require that... same as above, I speak from memory but I think there was one)

• District adjacency bonuses like +1 faith for every two adjacent forest (woods) tiles

And... that's kind of it. This is only considering your exact scenario.

For endgame scenarios, with expanded cities you should read this post: http://forums.civfanatics.com/threads/mines-or-lumbermill.391185/ - apparently in endgame lumbers become better than the mines (it is a civ 5 post but I think it still applies here as well, although a bit less since districts and placed wonders limit the possible space).