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.