A Lumbermill is great for mid to late game cities. The AI isn't always so quick to chop every tree, this means many captured cities will have trees around and if they're not riverside they're usually better as lumbermills for instant production. Also, Forests can exist on Plains and Grasslands, a Lumbermill there turns normally unproductive tiles into instant high production tiles.
What are you talking about? The AI normally clear cuts with abandon; unless you have a mod or are playing a difficulty with which I lack significant experience (< Emp or possibly Diety); the AI virtually ALWAYS clearcuts anything they can. Give the AI 70 turns with a tile, and I'm willing to bet the forest will be chopped.
Now if you rush early game or snag colonies, or something else atypical (i.e. play with a mod or an old patch) there
might be trees left, but the AI tends to clearcut if you give it enough time.
Regarding "Instant production"; you get the same production out of WS "instantly". If you are in SP, you get the same production and same food (which gives you 30 modded

as long as health doesn't matter). Caste, of course gives you more production still. If you have low pop, slavery and bio farms gives you more instant production. Lumber mills, sans health concerns, are only good before chemistry (but after RP) or after rails (but before SP or Biology). If you just want fast instant production, WS the place. You get the instant

(say to build production buildings). and you get the same or more effective

with farms or WS. If you want true "instant" production nothing beats slaving farms and perhaps drafting.
Mithradir's analysis for building a farm in a happy-capped city fails to take into account that you need two citizens to get one bio-farm and one specialist; so one specialist equals two grassland forest lumbermills worked.
Two lumbermills beat a specialist anywhere outside a GP farm (even with representation).
No it doesn't. Post biology you get +2

per farm; thus a bio farm will equilibrate you to one additional population than your lumbermill. There is some dynamic loss, but things equilibrate quickly unless you are at sufficiently high populations so that health is going to be a bigger concern anyways.
Two grassland forest LM gives you 4

and 6

as your terminal result. Two grassland farms (irrigated, post bio), gives you 8

; which with you can feed two specialists. If we have no one to work these two tiles, but start off at + 2

we do have to grow 2x as many people for rep specs, however we will grow ~1.5x as fast (after the first farm is worked we grow 2x as fast, then we grow 3x as fast with, then drop back to 2x as fast, and then are done; back end waiting drives this up to ~1.5). So yes there will be some dynamic loss, but under situations where health is not a concern. Note I am not talking about a city which has reached the happy cap, just a city where the happy cap is lower than the healthy cap so we can fully exploit these tiles however we wish without having to worry about health.
It is true that state property and caste system workshops outproduce lumbermills (+1H), but having to run SP denies you the access to corporations benefits. Non-SP, caste workshops are -1F +1H regards to lumbermills, and then I normally prefer the food (you need 3 grasslands to work 1 bio-farm and 2 workshops, giving 8H, while 3 lumbermills would give 9H). Without caste system your SP workshops are as good as lumbermills, and without either CS or SP you simply lose out. Remember moreover that running CS often becomes impossible in later stages of the game due to emancipation unhappiness.
No
really? Caste leads to emancipation unhappiness? I NEVER NOTICED THAT?
Look my point is not that lumbermills lose under all civic combinations when you ignore the health question; it is that they are sufficiently close (ignoring health) that you can safely assume that chopping out 20(30)

and compounding that for say 100 turns will net you more

in the long run. In the above example (no Caste, SP) it will take 30 turns to recoup the chop value of the forests (ignoring the fact that growth dynamics push this out just slightly further); with
no compounding. With compounding we are then looking at the relation: 3t*sum( [1+r]^n; n=0 to n = t-1) <=> 90 * (1+r)^t where t is the number of turns, and r is the rate of production growth per turn. For many values of r and t, it is a win to go for chopping and building crappy workshops. Building caste only WSs drives this ratio higher and SP only WSs dictate that you will never win to not chop (ignoring GW/health concerns).
If health isn't a concern, you should expect the magic of compound interest or relatively few turns you have to play to push the calculation over to chop netting you more effective hammers long term.
Finally cottages; they provide (as town) 7C1H running universal suffrage and free speech. Establishing a good transfer rate for commerce to hammers is a highly non-trivial task, but assuming we will actually use the extra commerce to rush-buy production, and we have maximal production and gold bonuses, both gold and production is doubled, so the rate is a factor 3 (without Kremlin). 7C would thus translate to 2 1/3 H, and a town would give 3 1/3H. However getting this extra bit of production requires a lot of conditions to be met: 2 civics, and the building of all commerce multipliers in a city, which might not really need them. Losing any of the gold/research multiplier buildings though reduce the value of the commerce, changing a civic either makes the transfer of gold to hammers impossible, or reduces the production of a tile to 5C1H = 2 2/3H<3H.
oh please. Firstly, 3 civs have UBs that push the rate higher. Secondly, every financial leader in the game also gets a better rate of return. Thirdly, commerce buildings (with the sole exception of the bank) provide additional benefits so you will almost invariably want them anyways (happy from markets, health from grocers, etc.). Fourthly, the last normal gold mod building comes at banking; hammer mods come last at assembly line (assuming power is trivial) so for a substantial portion of the game you can actually
build all the

modding buildings where you can't even build all the

modding ones (let alone the GW/unhealthy consequences of industrializing). Fifthly,

modding buildings also affect trade

so if you skimp on the buildings in general you are falling behind.
All of this ignores the central arguement made: that lumbermills are supposed to be more flexible; whereas in reality I get more from cottages and I have an obscene amount of flexibility to swap between production, science, cash, and culture. If I have the Kremlin, forget about it.
In short the lumbermill will be surpassed by different other improvements given the right conditions, but since its output is independent of any civics it is a pretty good tile in all conditions, while the contrary options might sometime be completely doomed (i.e. ever tried going out of state property?).
Yeah, half the time when I capture Sushi or Cereal. Lumbermills are ONLY good mid to late game. It is rare for them not to give you more net hammers in the long run by chopping stuff out (settlers, units, or improvements) and utilizing compound interest to end up better. The margins for going lumbermills are not good enough under any civics to outweigh the compounded return on the initial chop (and trait doubling/resource makes this is insanely obvious). It is only when health becomes a concern that lumbermills are useful. .4

is huge, and hence why lumbermills have their place.
Now I realize that I am not helping the OP by showing that lumbermills are actually pretty good in the late game. Indeed I would think that in almost all cases having the hammers earlier will provide a boost which will translate in a much bigger advantage than being able to work a lumbermill in the late game provides. However if you happen to have some forests left in your BFCs in the modern age, you should seriously consider lumbermilling them instead of chopping the forest and building another improvement instead.
Only out concern of health or GW. Under the best of scenarios for mills (no SP, caste, but RR), you need to play at least 30 turns (where production matters, as opposed to just simple mop-up) for 3 forested squares not to be better off as WSs and a farm. With a respectable rate of growth (say by more efficiently stealing a neighbor's land), you will never pay back the oppurtunity cost of not chopping.