Here is an attempt. I have called (wce/wh1)==Sqrt[a] and wpe^2/wh1^2==Sqrt[b] and manipulated the equations to express a as a function of b:

DivideSides[wce[B]/wh1[n, B] == Sqrt[a], 
  wpe[n]^2/wh1[n, B]^2 == Sqrt[b], GenerateConditions -> False];
ApplySides[#^2 &, %];
MultiplySides[%, 2 e^2 n^2/(B^2 e0^2), GenerateConditions -> False];
AddSides[%, -((B^2 e^2)/m^2 + (B^2 e^2)/M^2 + (e^2 n)/(e0 m) + (
     e^2 n)/(e0 M))];
ApplySides[#^2 &, %] // FullSimplify;
MultiplySides[%, 1/(1/(B m M) e), GenerateConditions -> False]
Solve[%, a] // FullSimplify

You can check the single steps by removing the semicolons. I don't know if this is what you need.

Solve the equation without giving numerical values to the variables:

Reduce[(wh[n, B] == wh1[n, B]) && e0 > 0 && 0 < m && 0 < M && B > 0 &&
   e > 0, Reals]

You will see that the only solution in the reals can be n == 0. Still, the two functions wh and wh1 are very close in numerical value:

values = {B -> 0.32, e -> 1.6*10^(-19),
   e0 -> 8.8*10^(-12),
   m -> 0.91*10^(-30),
   M -> 1.67*10^(-27)};
Plot[(wh[n, B] - wh1[n, B]) /. values,
 {n, 0, 1000000}, PlotRange -> .0001]