In a Manipulate like this,

Manipulate[Plot[T, {r, 0, R}],
 {{t, 15(*h*)*60*60}, 0, 24(*h*)*60*60},
 {{s, 0.0225}, 0, 0.05}]

the s and t variables need to be part of the expression in the body of the Manipulate. What you have in the body, Plot[T, {r, 0, R}], is an expression, it's not a predefined procedure that Mathematica will reference. Mathematica needs to "remember" that expression so that it can recalculate it when s or t change, so it can't just automatically compute it (which, if it did, would presumably expose the s and t as you seem to expect). So, you change s and Mathematica looks at Plot[T, {r, 0, R}] and says, "nothing to do here", because it doesn't see any s in that expression.

On a quick scan, I didn't even see where T was defined, but whatever the expression is that uses s and t, that's the expression you want to plot.

Also, you mention that T depends on t and s(t), but you set up the Manipulate with s and t as independent variables. I'm not sure what you expect to happen here. It seems to me that you should only need t. Whatever the function s is can be used in your definition of T by just applying it to t.

Mathematica needs to "remember" that expression so that it can recalculate it when s or t change, so it can't just automatically compute it [...] So, you change s and Mathematica looks at Plot[T, {r, 0, R}] and says, "nothing to do here", because it doesn't see any s in that expression.

The expression for T is defined few lines above the plot. Even though I have set values for s and t below that line, I understand that Manipulate should still treat T as an expression T=f(r,t,s(t)). And therefore it should change the numerical value of T if I input a different set of t and s(t). So s and t should exist in the body of Manipulate.