Does this scheme break under a DHP adversary? Consider the following cryptography scheme:

We are given a prime $p$. 
Alice wants to send the plaintext $m \pmod p$ to Bob.
Alice chooses $a$ s.t. $\text{gcd}(a,p-1)=1$, and sends Bob, $u = m^a \pmod p$.
Bob chooses $b$ s.t. $\text{gcd}(b,p-1)=1$, and sends Alice, $v = u^b \pmod p$.
Now Alice sends Bob, $w = v^{a^{-1}} \pmod p$, where $a^{-1}$ is the inverse of $a$ modulo $p-1$. Bob now calculates $w^{b^{-1}}$ to get back $m$.

Now I want to know if we have a passive adversary Eve who is able to solve the Diffie-Hellman Problem, gain access to $(u,v,w)$, can she get back $m$.
Now I want to know if we have a passive adversary Eve who is able to
  solve the Diffie-Hellman Problem, gain access to $(u,v,w)$, can she get
  back $m$.


Yes. But first you may want to note that this is the classic three-pass protocol, also called Shamir 3-pass protocol or Massey-Omura protocol which has been shown by Sakurai and Shizuya in 1998 to (sorry for the paywall) in general require a stronger assumption than (C/D)DH for security. 

But now for the attack, which is directly derived from the above security relation result. First note that $u=m^a,w=m^b,v=m^{ab}$ and that $p$ is known. So now suppose that you have an oracle $\mathcal O$ that given $(g^a,g^b,g,p)$ returns $g^{ab}$ that is a classic Computational Diffie-Hellman (DDH) Oracle. Now for a second, call $v=g$ and note that $u=g^{b^{-1}}$ and $w=g^{a^{-1}}$. Now feed $(u,w,v,p)$ to the oracle and get $g^{a^{-1}b^{-1}}=(m^{ab})^{a^{-1}b^{-1}}=m$ back. 

If you "only" have a Decisional Diffie-Hellman (DDH) oracle, note that you can still guess a message $m$, that is guess the return value of the above CDH oracle if there are only a few possible messages (not excluded in general in cryptography) and thus confirm your guess.