Tag light not working

My first thought is did the wiper work before you changed the battery.
The fact your windshield wipers work and nothing else makes it a headscratcher.
I think you will have to do the basics
Make sure your battery is good, Positive cables are good and negative to ground cables are good. If you have no power to fuse box or any fuses you could have bad battery cables or grounds.
I put a few links and the last one is a video.
Below I have owners manual with fuse box diagram, fuse box diagram for 09-12, electrical manual and video with some basic electrical checks. I think you need to start checking at the battery and work out from there to find where the power is lost.

https://www.mazdausa.com/siteassets/pdf/owners-optimized/2007/cx7/2007-cx-7-owners-manual.pdf

https://www.carknowledge.info/mazda-cx-7-2009-2012-fuse-box-diagram/

http://www.cheavens.com/Mazda/CX-7/CX-7%20WD.pdf

https://www.youtube.com/watch?v=DfLyh43iihM

I'd check battery connections, positive and negative. Check fusible links where applicable. Pick any component that doesn't work and that uses a fuse, check that fuse circuit for voltage with a test light. Some fuses are hot all the time, some go hot with the key on. If no voltage, we can check power distribution wiring diagram and go from there. Your owners manual should show which fuse does what. If you have no owners manual, fleet.ford.com has owners manuals, it's free to read online.
Just pick any system, hopefully a simple one, and I'll post a wiring diagram.

You can check the fuse, but the problem usually is the bulb or the socket contacts where the bulb plugs into.
Tag lights (license plate lights) are exposed to a lot of weather/road exposure and bumps/vibration, which shorten their life.

Rarely is a "tag light fuse" listed. The tag light is typically on a circuit with several other lights or accessories, and most commonly on the taillight circuit fuse, but not always.
Only a wiring diagram or a manual will be able to verify which circuit/fuse it is on, so you may have to check several fuses.

IF it turns out to be a blown fuse, your first step should then be to trace and identify the source of the electrical short (usually in wires by the tag light itself). Simply replacing a fuse is NOT a cure or a good 'repair'.

(As I write this, I have 2 tag lights out on my vehicle,, one which I verified is a blown bulb, and the other due to a badly corroded bulb socket)
Please rate via the helpful button if this helped answer your question, and please post back (via comments) when you discover the actual source of the problem.

Our fuse diagrams come from the owners manuals which are available online, these manuals only go back to the 2004 model year. I suggest you try eBay for a used owners manuals.

The fuse for the license plate light, if that is what you mean by tag light, is under the rear seat cushion On my 2005 deville it
is listed as the dimmer fuse. It is a 10 amp fuse. Look in your owners manual for its location.

No fuse? Or no slots? If 53 has little metal terminals inside the slots to accept a fuse, then fuse it! Someone pulled it out and put it into 52. If there was a previous owner who blew his heated seat fuse, maybe he traded his back-up lights for a cozy heated seat!
If, however, #53 has nothing there, like just an empty place of the fuse box-not wired, then indeed, strange. Is that the owner's manual you referred to? That should be accurate, if it's the owner's manual. Look closely at that #53
Good luck..

The diagram is part of the owners manual, you can access a free pdf copy HERE.

Go to autozone.com and register for free. Registration will give you access to manuals for your car, one of which will have a diagram of the fuse panel with an explanation of each fuse.

Hi, see page 6-57 of the owners manual: http://www.extendedgmwarranty.com/owners-manual/buick/1998-Buick-Century.pdf
The tag lamps are on fuse 17. Not sure if the reverse lights are on that same fuse or not, as they are not listed.

When I have no owners manual, and no fuse panel markers or guide or a sticker on the door, then I revert to using a 'fuse tester' available in fuse kits at places like Walmart or auto parts stores, that can help to diagnose blown out fuses.

Just connect the fuse tester to the fuse while the car's ignition is on, and it will light when the fuse is blown.

It only takes like 5 to 10 minutes to test all of them.

And is cheaper than acquiring the owner's manual.

If you give me the year, make (Ford), and model, I can check to see if the online manual has a copy and paste diagram of the fuse box. It varies from model to model whether I can do it.

Good luck on this repair and on getting the cig. lighter to work.