How did you arrive at the idea that "time is proportional to the square root of mass"? I imagine that you have some formula in which something, say a raindrop, is accumulating mass as it falls. In that case, I would be more inclined to say "mass is proportional to the square of time". In order for that to make any sense, your constant of proportionality would have to "fix" the units:
For example, if we have m= ct2, mass is measured in kg and time in seconds, then we have kg= c*sec2 so c must have units of kg/sec2 in order to "cancel" the sec2 and introduce kg into the formula.
Yes, you could also write t= c\sqrt{m} in which case c must have units of \frac{sec}{\sqrt{kg}}. While mathematics does not deal in "cause and effect", physics does. As soon as you introduce "mass" and "time", it makes more sense to me to think of time as the independent variable and mass as the dependent variable!
Notice that just because you can do a mathematical operation doesn't mean it will make physical sense! If A= x2, where A is the area of a square of side length x, then x= \sqrt{A}. Now, there, of course, A would have to be measured in something like "square inches" or "square meters" and, of course, x= \sqrt{A} would me measured in "inches" or "meters". That's why, in formulas, we typically write "square meters" as "m2" or "square inches" as in2.
On the other hand, I would not expect to have any physical quantity with units of \sqrt{m} or \sqrt{kg}. In the examples above, your constants (which may not be "physical" quantities) had to change the units appropriately
