So looking into a first film camera and I've always heard good things of Leica M cameras. I am familiar with the Nikon FM and its successors, but those cameras all contain an exposure meter. Has Nikon made a camera, say like the FM, that doesn't include an exposure meter?
There have been a few, these are the ones I know about --
(These are all, of course, film cameras)
The Nikon SP, a rangefinder camera not using the F mount
The Nikon F, the first F mount SLR from that company. The viewfinder/pentaprism was removable and while there was an optional viewfinder with an exposure meter but you didn't have to use one.
The Nikon F2 also had the exposure meter built into a removable viewfinder, but they made viewfinders without meters.
Note the FM and FM2 (my favorite) do have built in exposure meters -- but that is the only thing electronic about them. Pull the battery and the camera is still fully functional.
Cameras before about 1960 generally did not have light meters in them (there were only a very few in the very late 1950s). Those after 1960 mostly did. This corresponds to the introduction of semiconductors. The first Nikon F SLR was 1959, no meter. The Nikon FTn version added a meter in the mid-1960s. There were actual debates around this time about "can we trust a camera to do the metering?" :)
Leica M was a rangefinder, which is problematic for a builtin meter (what is the meter seeing to measure?) Because a meter has a width of field of view, often around 40 degrees, corresponding roughly to a "normal lens" view. So a general meter stuck on a rangefinder meters typically see about this 40 degrees, regardless of which lens is used, or what the film will see. However, a SLR with the meter behind the lens has advantage of metering the same angle as what the lens actually used sees.
But on any such camera, we can ignore the meter and just use Manual mode.
If you want a Nikon, I'd definitely go for the FM or FM2(n) and just not put a battery in. All the battery powers is the meter – without a battery the camera is completely manual.
If you want a rangefinder (since you mentioned Leica) but don't want to shell out $5k, there are cheap Russian Soviet-era Leica clones you can find. The Zorki 4 or 4k is commonly spoken about as "the best Leica clones you get", but there's also FED and Kiev models to look at. These early ones don't have light meters and are completely mechanical.
I recently went through that decision between Zorki and Nikon, and went with Zorki for its pocketability. (It's not small by any means, but the lens is shorter than regular SLR lenses, so it fits in the pocket of my jacket that way.) The Nikon FM cameras are small too though, and with an E-series prime lens they don't get too much bigger.
Nikon started to produce camera with rangefinders in 1948 (1, M, S, S2 etc...) and these ones does not have exposure metering (of my knowledge as I don't own any). Here is the rangefinders model list and some documentation here.
After they started with the professional reflex (SLR) Nikon serie F, F2, F3, etc... In this generation only the first models were without metering but sometimes optional like F and F2 with specific viewfinders.
The F and F2 without metering are the one equipped with the eyelevel finder.
In parallel, Nikon launched the Nikkormat models, more focused for the amateurs than the F series: only the first model of Nikkormat does not have exposure meter: the FS. All the other ones have it. Be careful when ou buy a used one, the metering could be damaged after so long years.
After all model from the FM serie (FM, FM-2, FM-3A and FM-10) have an exposure meter but they can also work without batteries, then without exposure metering.
The FE and FE-2 are electronic models and they need batteries for the shutter to be used at the speed value, then the metering is always present and active on these models.
This feels like you've got some embedded assumptions backwards.
Yes, early Leicas are lovely cameras (apparently, I've never used one). Yes, they made some good lenses, and you can still use them to take very nice photos.
No, they didn't all have built in light meters. For one thing, with a rangefinder it's harder to do through the lens metering so it's less accurate anyway.
These factors are (to the best of my knowledge) unconnected. An inbuilt light meter doesn't necessarily make the camera worse, or (except in a very few cases) even compel you to use it. And I'd recommend you use it in most cases, guessing isn't easy and external light meters aren't an improvement until you've got a very fancy one.
If you want to consider the significant difference between the systems - Leica are mostly famous for their M mount rangefinder cameras, with a smaller number of the less successful R mount SLRs. Nikon made a (relatively speaking) small number of S mount rangefinder cameras, then the much more successful F mount SLRs. The difference between a rangefinder Leica and an SLR Nikon is much, much greater in its impact on your shooting than whether your camera has a built-in lightmeter or you have to use a separate handheld meter.