So imagine there are now two dimensions of time, instead of the one dimension which we're used to. The laws of physics which we're used to won't make anymore sense unless they're adapted for the two dimensions of time. How would you adapt them?

Wear 2 wrist watches? Seriously, I cannot conceive of 2 time dimensions - I don't know at all what that would 'look like'.

Start with the basics. For example, an object would now have two velocities: one describing its rate of movement with respect to the first time dimension, and another one describing its rate of movement with respect to the second time dimension. That is: $v_1=dx/dt_1, v_2 = dx/dt_2$ We might also have to worry about how the rate of one time variable varied with respect to the other (i.e. what is $dt_1/dt_2$?). This is essentially DaveC's point: if the times were independent of one another then we'd have $dt_1/dt_2=0$, for example. The dynamics of an object with two dimensions could be formulated in many different ways, potentially. With our single familiar time our most basic laws of physics are constructed so as to be consistent with observation (experiment). If a new time dimension was discovered, we'd need to do experiments to start to formulate a consistent physics with the new dimension included. The point is: you can't derive what the universe is like from scratch just by using mathematics. The maths has to connect to experiments at some point to accurately model the real world. Relativity would very likely introduce some complications, too. As things stand, we know that space and time are not really separate, but rather form a unified construct that we call spacetime. If a second time dimension was discovered, that would probably require a fundamental reformulation of relativity, along with everything else.