To take advantage of being 20 ms early, you'd need to be able to detect neutrinos at greater than 50 Hz from the other side of the world. For context, the planned 40 kton DUNE far detector a mere 1300 km from the world's most powerful neutrino beam at Fermilab, will have a neutrino rate from the beam of a few a day. So yes, highly impractical.
You can also add modulation to the beam to encode more information to help discriminate against backgrounds but that might require even more rate...
With neutrinos that’d be a different story. They travel at the speed of light (or just below it - depends who you ask) and they aren't readily absorbed by anything. Therefore they can travel through the earth instead of around it.
Problem?
Generation, modulation and detection of neutrinos.
Not necessarily, if produced with an accelerator. (Potentially a nuisance to gamma-ray spectroscopists, generating spurious anisotropic scattering peaks.) Otherwise, yes.