It wasn't the politicians who fixed the Y2K bugs either. It was people like me.
It will be the same with Brexit - once the politicians have moved along with their time-wasting dance, it will be down to people who actually work for a living to make the necessary changes.
Some things certainly won't be complete by the target date - the politicians have already wasted over a year. There will have to be an interim period where many things carry on as at present.
Even if the politicians had got their act together (and in my opinion it's mainly the EU side that are delaying things) then it still doesn't make sense to try to change everything over on the same day - much better to plan a phased process where, say, regulations about medicine come into force on one day, and regulations about flights some other day.
Plan the most important and/or easily achievable changes to take place straight away and the less important or more difficult ones later. As with any large complex project you'll expect some time overruns on some issues and the sensible adult approach on those is to say, "We're not ready to make that change yet - so until we are we'll carry on doing things the way we have for the last several decades."
Has it occurred to you that the EU has far less to lose. May hasn't told the Conservative party what she wants, which is because the Tories can't agree. At the moment, for some reason, they can just about agree that May is better than Corbyn.
Our negotiating position seems to be "give us what we want... er once we have decided on it... or we'll cut off one of our feet and you don't want blood all over your nice suit."