There need not be any ordinal equipollent to $\Bbb R$, so there isn't really a standard notation for the least such (at least, none that I'm aware of). Note also that $\omega_1$ is typically used to denote the least uncountable ordinal, except when the reader is expected to have little to no familiarity with ordinals, in which case $\Omega$ is used instead. (Thanks, Brian, for prompting the clarification.)
More generally, we'll typically take $\omega_\alpha$ to be the least ordinal having the cardinal $\aleph_\alpha$. In fact, we'll take $\omega_\alpha$ and $\aleph_\alpha$ to be the same thing. We'll choose one notation over the other depending on whether we're talking about cardinalities or order types, or whether we're using cardinal arithmetic or ordinal arithmetic.
If we assume AC (or at least enough choice so that $\Bbb R$ is well-ordered), then the cardinal of $\Bbb R$--that is $\mathfrak{c}$--can be taken to be an ordinal, in particular the least ordinal equipollent to $\Bbb R$.