Giving your series a name, I'll set $$A(P,Q) \overset{\rm def}= \exp\bigl( \log(1 + P) + \log(1 + Q) \bigr) $$
to be your power series, where $P,Q$ are free (noncommuting) variables. I'm not sure what you want to know about this series: it exists, I don't think it has a name, the first few terms are
$$ A(P,Q) = 1 + P + Q + \frac12(PQ + QP) + \frac1{3! \cdot 2}\left(-P^2 Q + 2PQP - Q^2 P-PQ^2 + 2QPQ - Q P^2\right) + $$
$$ + \frac1{4!}\left( P^3 Q - P^2QP -PQP^2 + QP^3 - PQ^2P + PQPQ + \{P\leftrightarrow Q\} \right) + \dots.$$
I don't see much of a pattern in the coefficients, and I haven't worked out the next term. If the cubic term didn't have that $\frac12$, or if the quartic terms had more fractions, I would be happier. As it should, when $[P,Q] = 0$, the series truncates to $A(P,Q) = 1 + P + Q + PQ$. It is left-right symmetric and symmetric in $P\leftrightarrow Q$.

Here's one remark that might be useful for your intended application. Let $K = k\langle\langle X,Y\rangle\rangle$ be the ring of power series in two noncommuting variables. Then there is an algebra homomorphism $\Delta: K \to K \hat\otimes K$, where $\hat\otimes$ denotes that you should complete the tensor product w.r.t. the adic topology, given on generators by $\Delta(X) = 1 \otimes X + X\otimes 1$ and $\Delta(Y) = 1 \otimes Y + Y \otimes 1$. Recall that an element $P \in K$ is *primitive* if $\Delta(P) = 1 \otimes P + P\otimes 1$. Then the primitive elements form a Lie subalgebra of $K$, and consist precisely of the *Lie series*: the power series that can be expressed without ever referring to multiplication in $K$, only to the Lie bracket $[x,y] = xy - yx$ (and that begin in degree $1$). Recall also that an element $P\in K$ is *grouplike* if $\Delta(P) = P\otimes P$. Then the grouplike elements if $K$ form a group under multiplication; in particular, they are all units.

If $P,Q$ are primitive, then there's no particular reason for $A(P,Q)$ to be primitive, and actually I think it never will be. However, by construction $\Delta$ is continuous w.r.t. the adic topology, and it is a homomorphism, and so $\Delta(f(P)) = f(\Delta(P))$ for any power series $f$ in one variable. Also, an element $P\in K$ is primitive iff $\exp(P)$ is grouplike. So it follows that if $1+P$ and $1+Q$ are both grouplike, then so is $A(P,Q)$.

I'll close by saying that, to me anyway, you already have an "explicit expression" for the power series, and even a "geometric interpretation", which is that it moves the additive structure of the Lie algebra to the (formal) group (whereas the BCH series moves the group multiplication to the Lie algebra). You shouldn't strongly hope for a simple description of the coefficients in terms of combinatorics, because similar descriptions for BCH, although they do exist, can be rather complicated.

**Update**

Above I observed that $A(P,Q)$ is not a Lie series in $P,Q$. This can be seen directly: any Lie series truncates to its linear and constant terms when $[P,Q] = 0$, whereas $A(P,Q) = (1+P)(1+Q) = 1 + P + Q + PQ$ in commuting land.

So the next best thing, as I asked in the comments, is whether
$$A(P,Q) - (1+P)(1+Q) = -\frac12[P,Q] - \frac1{12} \bigl( [P,[P,Q]] + [Q,[Q,P]] \bigr) + \dots $$
is a Lie series. Alas, this also fails, as can be seen at the quartic part. Indeed, by Jacobi and antisymmetry,
$$ [P,[Q,[P,Q]]] = [[P,Q],[P,Q]] + [Q,[P,[P,Q]] = [Q,[P,[P,Q]] = -[Q,[P,[Q,P]] $$
is the unique Lie monomial (up to scalar) of degree $P^2Q^2$. But it is antisymmetric under $P\leftrightarrow Q$, whereas $A(P,Q)$ is symmetric under the same transposition. Thus if $A(P,Q) - (1+P)(1+Q)$ were a Lie series, then it could not have any terms of degree $P^2Q^2$, whereas by direct calculation:
$$ A(P,Q) \ni \frac1{4!} \bigl( PQPQ - PQ^2P - QP^2Q + QPQP \bigr) = \frac1{24} [P,Q] ^2 $$