Hmm a good nerd-snipe puzzle. I was never very good at physics, so hopefully someone can check my work... assuming upon mixing coffee is at Tc and milk at Tm, and simplifying to assume equivalent mass & specific temp we have (Tf - Tc) = -(Tf - Tm) => Tf = (Tc+Tm)/2 which is intuitive (upon mixing we get the average temperature).
On the assumption that the cold milk is always at a fixed temperature until it's mixed in, then the temperature of coffee at point of mixing is the main factor. Before and after, it follows newton's law of cooling. So we're comparing something like Tenv + [(Tc+Tm)/2 - Tenv]e^(-2) vs (Tenv + [Tc - Tenv]e^(-2) + Tm)/2. The latter is greater than the former only when Tm > Tenv (the milk isn't cold), or in other words it's better to let the coffee cool as much as possible before mixing assuming the milk is colder than the environment.
Another interesting twist is to consider the case where the milk isn't kept at a fixed temperature but is also subject to warming (it's taken out of the fridge). Then the former equation is unchanged but the latter becomes (Tenv + [Tc - Tenv]e^(-2) + Tenv + [Tm - Tenv]e^(-2))/2. But this is equivalent to the former equation, so in this case it doesn't matter when you mix it.
Not 100% confident in both analysis, but I wonder if there's a more intuitive way to see it. I also don't know if deviating from the assumption of equivalent mass & specific temp changes the analysis (it might lead to a small range where for the fixed case, situation 1 is better?) It's definitely not "intuitive" to me.
On the assumption that the cold milk is always at a fixed temperature until it's mixed in, then the temperature of coffee at point of mixing is the main factor. Before and after, it follows newton's law of cooling. So we're comparing something like Tenv + [(Tc+Tm)/2 - Tenv]e^(-2) vs (Tenv + [Tc - Tenv]e^(-2) + Tm)/2. The latter is greater than the former only when Tm > Tenv (the milk isn't cold), or in other words it's better to let the coffee cool as much as possible before mixing assuming the milk is colder than the environment.
Another interesting twist is to consider the case where the milk isn't kept at a fixed temperature but is also subject to warming (it's taken out of the fridge). Then the former equation is unchanged but the latter becomes (Tenv + [Tc - Tenv]e^(-2) + Tenv + [Tm - Tenv]e^(-2))/2. But this is equivalent to the former equation, so in this case it doesn't matter when you mix it.
Not 100% confident in both analysis, but I wonder if there's a more intuitive way to see it. I also don't know if deviating from the assumption of equivalent mass & specific temp changes the analysis (it might lead to a small range where for the fixed case, situation 1 is better?) It's definitely not "intuitive" to me.