For $n\in\mathbb{Z}_{\geq 0}$ and $0\leq i< 2^n$, denote

$$
X_{n,i}=\sup_{t\in[i2^{-n}, (i+1)2^{-n}]}{|W_t-W_{i2^n}|}.
$$

Let $n$ be such that $2^{-{n}}<|\Delta t|\leq 2^{-{n+1}}$. Then,
$$\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\leq 4\sup_{i}|X_{i,n}|^p,$$ because such $s$ and $t$ must belong to a union of three consecutive diadic intervals of length $2^{-n}$. For a fixed $n$, $X_{n,i}$ are i. i. d. random variables; moreover, by Brownian scaling,
$$
X_{n,i}\stackrel{\mathcal{D}}{=}2^{-n/2}X_{0,0}.
$$
If $F(t)$ denotes the cumulative disctribution function of $X_{0,0}$, then we have, with $N=2^n,$
$$\mathbb{P}(\sup_{i}|X_{i,n}|\leq t)=F(N^{1/2} t)^{N}.$$
It is well known that $\mathbb{P}(\sup_{[0,1]} W_t>a)=2\mathbb{P}(W_t>a)\leq 2e^{-a^2/2}$ for $a$ large enough, so that
$F(t)\geq 1-4e^{-t^2/2}$, and
$$
\mathbb{P}(\sup_{i}|X_{i,n}|>t)\leq 1-(1-4e^{-Nt^2/2})^{N}.
$$
Let $T_n:=\sqrt{2\log 2}n^\frac12 N^{-\frac12}$, then, plugging into the above,
$$
\mathbb{P}(\sup_{i}|X_{i,n}|>kT_n)\leq 1-e^{N\log(1-4\cdot 2^{-k^2n})}\leq 10\cdot 2^{(1-k^2)n}.
$$
That is to say, the median of $\sup_{i}|X_{i,n}|$ is of order at most $T_n$ and its tail decays very sharply after that, implying

$$\mathbb{E}\sup_{i}|X_{i,n}|^p=O(T_n^p)$$ for all $p$, or
$$\mathbb{E}\left(\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\right)=O\left((|\Delta t\log (\Delta t)|^\frac{p}{2}\right).$$

These estimates don't lose much and they are of course very similar to Lévy's theorem on the modulus of continuity for Brownian motion.

On the Moments of the Modulus of Continuity of Itô Processesby M. Fischer and G. Nappo should either contain an answer or a reference to the right paper. $\endgroup$