In the absence of CH, there are just $2^{2^{\aleph_0}}$ ultrapowers of $\mathbb{R}$ of length $\omega$ up to isomorphism.

Question

We consider structures of the language of ordered fields. In the absence of CH, there are just $2^{2^{\aleph_0}}$ ultrapowers of $\mathbb{R}$ of length $\omega$ up to isomorphism. This seems to be a corollary of the following paper:

Linus Kramer, Saharon Shelah, Katrin Tent, Simon Thomas "Asymptotic cones of finitely presented groups" arXiv:math/0306420 [math.GT] https://arxiv.org/abs/math/0306420

But the paper deals with a more general setting, which is unfamiliar to me. Can a easier argument prove the first statement?