The amount of production of a great engineer on normal speed is calculated using the formula
300 + 30 * city population
So in a city with a population of 1 the engineer only results in 330 production, in a size 15 it results in 750 production.
For quick speed the formula is:
200 + 20 * city population
For epic speed it is:
450 + 45 * city population
and for marathon it is:
900 + 90 * city population
The amount of hammers does not seem to be influenced by era, map size or difficulty level.
Conclusion:
If you want to hurry late game wonders, build it in a city with a large population.
Let me rephrase the question:
When are Great Person tile improvement bonuses worth it?
Basically the rule of the thumb for all of them is the earlier you build the improvement the more profit it would bring during the course of the game. This is especially noticeable when you play for babylonians and get a Great Scientist from Writing. Such an early academy would increase your science output about two times and would really help to zoom out those early technologies.
Generally I like to build Academies. Let's do some calculations. Before University it gives you 5 science. Technologies till Education cost from 35 to 440 beakers, so roughly 237.5 in average. So your Academy would pay off in about 47.5 turns and later it would be just plain profit. So I think that's just great. Technologies from Education to Scientific Theory cost from 440 to 1300 beakers, so 870 in average. Academy with University gives you 7.5 beakers, so it would pay off in 116 turns, so that is still quite good. And if a city is dedicated to science and has a National College, then Academy would give you 10 beakers, so would pay off in 87 turns, which is even better. Later technologies seem very expensive and definitely worth using Great Scientist to discover. But the problem with them is that in modern era they almost stop increasing in value. And you obviously want to use your Great Scientist to discover later and more expensive technologies. So it kind of stops me from doing so when I see that 3rd tier modern technology would take me just 1 or 2 more turns to discover then 1st tier modern one. So IMHO you can get the best deal from your GC discovering late renaissance / industrial era technologies.
Another thing is then you are beelining tech tree to a specific unit, wonder or even era (for opening a SP branch). In that case its definitily worth using GC for instant discoveries.
I also almost always use Great Engineers for rushing wonders, but let's do some maths anyway. Lets say your best city before Steam Power could grow from 5 to 15 in size, so (very) roughly 10 in average. So rushing a wonder gives you about 300+10*30=600 hammers. A Manufactory without Factory would give you 5 hammers, so would pay off in about 120 turns. If you build buildings a Manufactory with workshop would give you 7.5 hammers, so would pay off in 80 turns. Actually its not such a bad deal when you think about it. So if you don't have any important wonders in sight it might just be worth building a Manufactory. The problem is that you always do :).
Using a Great Merchant for a trade mission would give you from 350 to 650 gold depending on era, so 500 in average. Add to that influence worth about 250 gold (I think you get about 30 points of influence, but I might be wrong) and you have 750 gold from a trade mission. A Customs House gives you 5 gold with no buildings and about 12 gold with stock exchange, so its 8.5 gold in average, which would make it to pay off in about 88 turns. Not such a bad deal. I'd say use it for trade missions when you desperately need some cash (lets say for unit upgrades) or CS influence. The economy of using Great Merchant for Golden Age inflicted gold output is explained in this answer.
The last two Great People are mostly situational. I only used Great Artist once ever for culture bombing when I needed access to coal and couldn't settle another city as I was going for culture victory. It might well be used to access some valuable resource tile, which is on that useless one tile piece of land off the coast more than 3 tiles away from a city. It might also be used in some stalemate war situation, when pushing a cultural front line just one tile is such a big deal, I've seen AI doing that once. In other cases I find it more effective to settle that land, take it by force, or if it is CS land - just ally with it.
I really like using Great Generals for building Citadels. Just put it in front of the city (for friendly lands and easy unit exchange), build a road to it, make sure there is enough room for enemy units to surround you citadel on other 3 sides (for -3 damage) and you can pretty much fend off any attack. In one game I was controlling about 1/4 of the pangaea map, and Mr Askia was controlling the rest, we were at never ending war and he was sending swarms of units at me. I successfully held the mountain range position using just several units, several forts and a citadel.
I usually use Great People for Golden Ages when I "need them". That is if I need a cash injection for unit upgrades or to bribe some CS's. Or if I'm falling behind in producing buildings. I also like to zoom through the phase of building workshops and factories using golden age, as the faster you get those buildings the more production you will have overall.
Best Answer
In general I agree - in late-game, GP are less useful and I usually trade them in immediately for a quick golden age. However, unless you have absolutely everything, great scientists and great engineers do have some value, and there are use-cases where the rest are useful as well, I've listed them below.
Remember golden ages are always great, even if they are short. +gold, +production and no happiness issues.
Great Scientist: if you already have everything then this is indeed useless, but as long as you don't this is still useful. It's true late-game technologies are usually researched quickly but there are a few wonders in late game and being the first to reach them can be nice; furthermore, being the first to unlock Aluminum or Uranium is good because you can plan your land acquisitions better.
Great Engineer: there are some wonders in the late game, and they are pretty good actually. Other than that, great engineers are useful if you create a new city in the late-game; I usually buy a lot of buildings whenever I do this, but a great engineer can save a lot of money by speeding the most expensive buildings.
Additionally, factories are always useful.
Great Merchant: whether you should use the diplomatic mission or the golden age is not always obvious - see my answer to another question to see my opinion. In general I say a diplomatic mission is usually worth more gold, but a golden age also nets production. Custom houses are not worth it, in my opinion.
Great General: I almost always trade them for a golden age immediately, with one exception - I sometimes build citadels in natural chokes or other appropriate locations. From experience, a single citadel in the right spot, backed by 2-4 units, can stop an army.
Great Artist: the most useful type in the late game! Using the culture bombs allows you to reach practically everywhere. I one time used 3 of them to get to just one Aluminum patch in the middle of an icy area (just took a little time because of the cooldown). They can also be used to steal stuff from civilizations and city-states.
Also, landmarks are always nice in cities that already have a lot of +% culture.
Regarding city states: I usually do use great people from city-states, late-game maintenance costs are high but I think a golden age is always worth more than the time it takes them to get to the nearest border, even if it's a golden age lasting just 3 turns.
EDIT as of the June 2011 patch, great person improvements have been buffed:
This means using great people for improvements is a more viable option now.