Some countries have already run trials, it is a delicate subject and blockchain technology as any other new technology must be tested and since voting doesn't happen that often, I believe it will take some time before we see it as a reality. Check these articles about the trial in Switzerland and the plans in Japan to do so:
In practice, the main issue will be the proof of identity, because even when the national id number can be use to identify the voter in the blockchain, there is no guarantee it will be used by the real citizen. If polls happen in a typical way, going to a physical place, then a standard procedure may be used to id a person, representative will check the official valid ID and then the voter can scan it in a device and vote on a screen instead of paper; if the polls are run on devices, such as personal computers or smartphones, some sort of biometrics recognition system must be used.
And then there is the politicians will... I believe all of them cheat, so, to authorize this technology is like shooting in their own foot.