Here is one reason not to expect such a relationship (although I'm not sure if it can be completed to a proof). The Jones polynomial $J_\sigma$ (roughly) comes from taking the trace of a linear map $A_\sigma$ associated to the braid $\sigma$, so the question (roughly) asks about relations between $Tr(A_\sigma)$, $Tr(A_\tau)$, and $Tr(A_\sigma A_\tau)$.

Let $a,b$ generate a free group $\pi$, and let $Chr$ be the $SL_2$ character scheme of $\pi$ (i.e. the space of pairs of $SL_2$ matrices considered up to simultaneous conjugation). Define $Tr(g):Chr \to \mathbb C$ to be the function $\rho \mapsto Tr(\rho(g))$. Then the functions $Tr(a)$, $Tr(b)$, and $Tr(ab)$ are algebraically independent. So there shouldn't be any universal formula relating these three traces, whether $a,b \in SL_2$ or are bigger matrices.