I read on a website that cheddar can be aged for 12 years
I suspect that 12 year old cheddar is so strong that a small amount may be equivalent in flavor to a much larger amount of mild cheddar (3 months old)
Assuming that all ages of cheddar are made the same way, how much of each one from 1 year old to 12 years old would be equivalent in flavor to 1 cup of mild cheddar in flavour?
Cheese isn't just used for flavor; it generally contributes greatly to the texture, whether via a crispy browned layer on top, or thick goopy texture inside a dish, or even just background richness. If you change the total amount of cheese, even if you have the same amount of cheddar flavor, you won't have the same dish.
So I would suggest that if you like cheddar, just use the quantity that's called for, and replace mild with medium or sharp. You'll get more flavor, and you'll probably like it.
Otherwise, if you want to keep the flavor of the dish roughly the same intensity but replace mild with sharp, then make up the rest with another mild cheese. Obviously it won't be the same flavor but it'll be roughly similar strength; if all you have is sharp cheddar, this will be better than using all sharp cheddar (too strong a flavor) or using half as much (upsetting the balance of the dish). For example, you might replace a cup of mild cheddar with half a cup of sharp cheddar and half a cup of mozzarella. (See rumtscho's answer for a good explanation of exactly how not the same it will be.)
But the ratio depends on the particular cheddar; you can get a pretty good variety of things marketed as "sharp cheddar". I think the half and half ratio will be a good starting point for nice strong, sharp cheddar, and I personally would just use medium instead of mild without changing any quantities. So experiment with the cheese you have and see what you like.
And remember, it'll never be the same, it'll just keep the sharp flavor from being overpowering. I'm not saying this is an exact substitute, just what I'd do if I only had sharp cheddar around.
Finally if you're asking this because you're planning on making and aging cheddar yourself... don't get ahead of yourself. See how well you do at the basic cheesemaking before you worry about aging it.
The conversion ratio you are asking for doesn't really exist.
There are only five tastes we taste with our tongue: salty, sweet, bitter, sour and umami. They are indeed different by concentration. But the human tongue response is not linear. When a source says that saccharine is 500 times sweeter than sugar, they mean that, if you dissolve 1 g of saccharine in a liter of water, a spoon of the solution if will taste as sweet as if you had dissolved 500 g of sugar. But if you put a gram of saccharine in your mouth, you won't say "wow, this is 500 times sweeter than sugar" - you will notice that it is very much sweeter, but not by how much.
Umami taste does increase in cheese with aging (and saltiness increases too, due to the loss of water), but we don't dissolve cheese, so we cannot really make a statement about this difference.
But even if we could find a number for the concentration ratio of umami, it wouldn't even be relevant. Most of the difference in taste between young and old cheese comes from ripening. In ripening, chemical and biological reactions produce completely new molecules in the old cheese which were not present in the young cheese. These new molecules have their own aroma, which is perceived as part of taste (although not perceived with the tongue). And it is not a matter of the young cheese having less of them; the young cheese doesn't have them at all, because the reactions which create them take so much time. So, old cheddar doesn't taste "the same only more so" than young cheddar, it tastes "differently" than young cheddar.
Asking your question is like asking "how many acorns do I need to get as many teak leaves as a grown teak tree". It just doesn't make sense. There are no leaves in the acorn yet; similarly, there is no ripe cheddar flavor in the young cheddar yet.
This is of course simplifying, because at some point, you start getting the flavor molecules which will be abundant in old cheddar, only you have less of them at the beginning. But even taking that into account, you cannot calculate a ratio because 1) not all molecule types which will be present in old cheddar are already present in young cheddar in perceptible amounts, and 2) the flavor profile - the ratios between kinds of flavor molecules - is completely different at different ages.
So, let's make a thought experiment. We take 1 kg of just-created cheddar, grate it finely, and start adding 12 years old cheddar in small amounts (basically diluting the old cheddar). If at X gram of old cheddar, this mixture tastes exactly like 1-year old cheddar, then the answer to your question would be X:100. But there is no such amount; no matter how much old cheddar you add or don't add, the mixture will never taste like 1-year-old cheddar. So, your question is not answerable. Cheese just doesn't work that way. Not even in flavor; Jefromi also explained why it doesn't work that way in other important cheese properties, and TFD explained that, if it did work that way, natural variance would make the ratio impossible to calculate.
This is probably an impossible to answer question, as every cheese batch is different, and how do you compare 6 year old cheese to 12 year old cheese of the same batch? Freezing cheese to stop aging, makes it very change in a different way
Cheese ages in a inverse exponential way, just line wine. It takes progressively longer and longer for flavours to change
Also, cheeses don't change in one direction of flavour, and over time they become quite different from where they started, so 12 year old cheese may have no relation to 2 year old cheese of the same type
Over long times the texture changes to the point of not being what people think of cheese is anymore, too more of a paste
Most people looking for aged cheese are perfectly happy in the 18 months to 3 years bracket, after that it gets a little weird
Remember cheese is aged at around 4°C, which for most people requires electrical refrigeration, which is therefore more expensive for every year for not much gain. You can't just shove some cheese in the cellar and hope it turns out OK
