Explicit surjectivity of Galois representations of products of elliptic curves over function fields

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for elliptic curves by Griffon--Pazuki, and techniques originated by Serre and Masser--Wüstholz in the number field setting. We apply our result to prove that most members of a family of products of elliptic curves over Q with no extra endomorphisms have no exceptional primes above a specified constant which depends neither on the elliptic curve factors nor on the dimension of the product.