I never did understand why people worry about making a symbolic tool give a result in a mathematically identical form, that happens to look as you want to write it. Can you spend an hour figuring out how to produce a form that looks as you want? Probably. Why?
You want to solve this essentially in a non-dimensional form for v. So write the equation by dividing by c^2. We will use the transformation u = v/c.
eq1 = (t)^2 == (tau)^2+(u*t)^2
Having done that, now undo the transformation, but do NOT use symplify which may screw things up.
Note that I had to use solvedeq1(2) to give the positive solution.