Let $F_i$, $i\ge 1$, be a sequence of finite groups such that for every prime $p$ the orders of all but finitely many groups $F_i$ are not divisible by $p$. Then your group $W/U$ has no homomorphism onto a nontrivial finite group. Indeed if $G$ is a finite nontrival homomorphic image and $\phi: W/U\to G$ is a homomorphism, $n=|G|$, $1\ne a\in G$, $b\in W/U, \phi(b)=a$ then $b=c^n$ for some $c\in W/U$, therefore $a=\phi(c)^n=1$ in $G$, a contradiction. In particular $W/U$ is not residually finite. I used the fact that if a prime $p$ does not divide the order of a finite group $F$ then every element of $F$ has a root of degree $p$.

Let $F_i$, $i\ge 1$, be a sequence of finite groups such that for every prime $p$ the orders of all but finitely many groups $F_i$ are not divisible by $p$. Then your group $W/U$ has no homomorphism onto a nontrivial finite group. Indeed if $G$ is a finite nontrival homomorphic image and $\phi: W/U\to G$ is a homomorphism, $n=|G|$, $1\ne a\in G$, $b\in W/U, \phi(b)=a$ then $b=c^n$ for some $c\in W/U$, therefore $a=\phi(c)^n=1$ in $G$, a contradiction. In particular $W/U$ is not residually finite. I used the fact that if a prime $p$ does not divide the order of a finite group $F$ then every element of $F$ has a root of degree $p$. Thus every element of $W$ has a root of degree $p$ modulo $U$.