What Nikon camera and lenses will be good for newborn photography?

I'm not a professional just a regular person with a little experience in photography. I'm interested in newborn and family photography for a side thing. Photography has always been a passion of mine. I learned to shoot on a Nikon and I'm interested in finding out what are some good lenses to capture the small details. I'm not looking to spend a whole lot of money as I'm just learning and beginning.

Would a D5200 be sufficient? And what lenses?

Thank you! Do you think I should also purchase a macro lens? The ceiling will be white as well.

The D5200 is a very good choice. The 18-55mm AF-S VR kit lens should be fine. The 18-140mm AF-S VR has nice versatility and is reputed to be sharp, but it has a lot of pincushion distortion.
The next thing to get would be the SB700 flash to do bounce flash for indoor shots.
I hope that your ceiling is white or nearly white.

Cheaper would be a Nikon D3100 with 50mm 1.8 lens.

The model Nikon you choose is your business, but I would definitely recommend a 70mm lens which is like 105mm on a 35mm film camera. This would allow you to back up a litte if you use flash. Plus if you use a Vello Light Bounce attached to your bounceable flash the light will be much softer and easy on newborn eyes.
A macro lens is not suited for this type of photography, best for inanimate (not alive) subjects.

Yes a D5200 would do you fine with the kit lens to start with. If you want faster lenses, the 50mm f1.8 AF-S lens is good for portraits, or if you want to back off a bit, the 85mm f1.8 AF-S.

You need AF-S lenses because the D5200 doesn't have a focus motor in the body (you can use AF lenses, which are cheaper, but you'd need to manually focus them).

You are also likely to need some off camera lighting - a small manual speedlight, softbox & stand can be put together for about £50.

Incidientally - you say that you are doing this as a 'side thing'. Are you/will you be charging money? If so, that puts a whoel different legal spin on it and classifies you as a professional photographer. You will need robust contracts and public liability insurance in order to minimise any possible legal proceedings if something goes wrong with a shoot.