Civilization – How to keep up on Science on Immortal

First of all, you can just sell them if you don't need them. Not only do you immediately get some gold, you also decrease the number of units you own, and units cost gold each turn.

Secondly, connecting a city to the capital by railroads greatly increases a city's production, so if you can afford it it's recommended.

Finally, you can replace improvements you don't need / want with other improvements. For instance, for cities that are big enough it's probably better to replace farms with mines or trading posts.

You can set your workers to "automate", and they will automatically build roads, replace them with railroads if available, and will replace improvements with other improvements the AI thinks you need more.

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.