Here is some information on the history of the result, which was actually proven before Shirota's 1952 theorem. It was proved in 1948 by Marczewski and Sikorski as Theorem VI in:

Marczewski, E.; Sikorski, R., Measures in non-separable metric spaces, Colloq. Math. 1, 133-139 (1948). ZBL0037.32201.

Their statement is that if $X$ is metrizable and $|X|$ is less than the first measurable cardinal, then every 2-valued Borel measure on $X$ is a Dirac measure (of some point or other in $X$). The converse is not directly stated, but is trivial - if $|X|$ is greater than or equal to the first measurable cardinal, it admits a countably-additive measure $\mu : \mathcal{P}(X) \rightarrow \{0,1\}$ vanishing on singletons, which can be restricted to a Borel measure, which, since it vanishes on singletons, is not the Dirac measure of any point.

Of course, to relate this to the usual definition of realcompact space, you need to use Hewitt's Theorem 16 from:

Hewitt, Edwin, Linear functionals on spaces of continuous functions, Fundam. Math. 37, 161-189 (1950). ZBL0040.06401.

which states that a completely regular space is realcompact iff every 2-valued Baire measure on it is a Dirac measure. (On a metrizable space, the Baire and Borel sets (in the relevant sense) are the same because for every closed set there is a real-valued function vanishing exactly on that set.)

Beware that back then they used an older terminology for measurable cardinals, where a cardinal $\kappa$ has "two-valued measure zero" iff it admits no countably-additive measure $\mu : \mathcal{P}(\kappa) \rightarrow \{0,1\}$ vanishing on singletons (equivalently, strictly smaller than the first measurable cardinal) otherwise it is "two-valued measurable" (in modern terminology, greater than or equal to the first measurable cardinal). To avoid confusion, no new publications should use the old terminology, even though it really is countably-additive measures that matter here (so one doesn't care about measurable cardinals larger than the first).

Here is some information on the history of the result, which was actually proven before Shirota's 1952 theorem. It was proved in 1948 by Marczewski and Sikorski as Theorem VI in:

Marczewski, E.; Sikorski, R., Measures in non-separable metric spaces, Colloq. Math. 1, 133-139 (1948). ZBL0037.32201.

Their statement is that if $X$ is metrizable and $|X|$ is less than the first measurable cardinal, then every 2-valued Borel measure on $X$ is a Dirac measure (of some point or other in $X$). The converse is not directly stated, but is trivial - if $|X|$ is greater than or equal to the first measurable cardinal, it admits a countably-additive measure $\mu : \mathcal{P}(X) \rightarrow \{0,1\}$ vanishing on singletons, which can be restricted to a Borel measure, which, since it vanishes on singletons, is not the Dirac measure of any point.

Of course, to relate this to the usual definition of realcompact space, you need to use Hewitt's Theorem 16 from:

Hewitt, Edwin, Linear functionals on spaces of continuous functions, Fundam. Math. 37, 161-189 (1950). ZBL0040.06401.

which states that a completely regular space is realcompact iff every 2-valued Baire measure on it is a Dirac measure. (On a metrizable space, the Baire and Borel sets (in the relevant sense) are the same because for every closed set there is a real-valued continuous function vanishing exactly on that set.)

Beware that back then they used an older terminology for measurable cardinals, where a cardinal $\kappa$ has "two-valued measure zero" iff it admits no countably-additive measure $\mu : \mathcal{P}(\kappa) \rightarrow \{0,1\}$ vanishing on singletons (equivalently, strictly smaller than the first measurable cardinal) otherwise it is "two-valued measurable" (in modern terminology, greater than or equal to the first measurable cardinal). To avoid confusion, no new publications should use the old terminology, even though it really is countably-additive measures that matter here (so one doesn't care about measurable cardinals larger than the first).