 Solution

For every positive real $r$, a centered normal variable of variance $t-s$ has absolute $r$th moment $c_r|t-s{|}^{r/2}$, where $c_r=\mathbb{E}|N(0,1){|}^r<\infty$. Fix $0<\alpha <1/2$ and choose $r$ large enough that $\alpha <1/2-1/r$. ThenThe Kolmogorov continuity theorem gives a modification with Hölder continuity of every exponent less than $(r/2-1)/r=1/2-1/r$, in particular exponent $\alpha$, on each compact interval. This modification and the given continuous Brownian motion agree almost surely at all rational times; continuity makes them agree everywhere simultaneously. Thus the property holds for the given trajectories, not just for another version.

Take a countable increasing sequence of positive exponents approaching $1/2$, and intersect their probability-one events for every integer time horizon. Hölder continuity of a larger exponent implies that of every smaller positive exponent on a compact interval, after adjusting the constant. We have therefore proved the simultaneous assertionThe compact-interval qualification matters: a single uniform Hölder constant on the entire half-line is not being asserted.