We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of $\mathbb{Z}^d$, $d\geq 5$. In particular, we prove that every tree in the forest has spectral dimension $4/3$ and walk dimension $3$ almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are $1$ and $1/2$ respectively. (The past of a vertex in the uniform spanning forest is the finite component that is disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is $2$ on any transitive graph of at least five dimensional polynomial growth, and is $1$ on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are $2$ and $1/2$ respectively for any transitive graph with polynomial growth of dimension at least five, and are $1$ and $1/2$ respectively for any bounded degree nonamenable graph. In the case of $\mathbb{Z}^d$, $d\geq 5$, some of our results regarding critical exponents recover earlier results of Bhupatiraju, Hanson, and Járai (2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work.