Example of a C*-algebra whose $K_1$ is uncountable

There must be tons of ways to do this, but a simple one is to start with an uncountable set $X$, equipped with the discrete topology, and consider $c_0(X)$. There are uncountably many pairwise inequivalent minimal projections in this algebra, so its $K_0$ group is uncountable. Now use $K_0(c_0(X)) \cong K_1(Sc_0(X))$ where $SA$ is the suspension of $A$.

Nik's answer nails it but if you prefer something representable on a separable Hilbert space then you may consider the suspension $SM$ of any ${\rm II}_1$-factor $M$. Indeed, as $M$ is tracial, $K_0(M) \cong \mathbb R$ and the suspension simply reverses the $K$-groups.

Actually you may produce further commutative examples that are representable on separable Hilbert spaces: for example $S\ell_\infty$, since $K_0(\ell_\infty)$ comprises $\mathbb{Z}$-valued continuous functions on $\beta \mathbb N$.