We will be in Madrid during this time and planned to go to Segovia and Avila that day on a tour. Will many things be closed that day and if we decide to stay in Madrid that day, how should we spend our time? Will Prado be open? Cathedrals?
For individual museums, you can check the website for days closed. That is usually reliable, though I suppose things might be a bit different this year. It may be more difficult to find online information for churches--at least it has been for me.
Oct 12th (Fiesta Nacional de España), is a public holiday, though hours for major museums, churches, and tourist sites should not be affected, you may find an uptick of visitors to these sites on the 12th, as many locals have the day off. Public transport runs normally. Outside of the touristy zones you may find some smaller shops/restaurants closed, but not too many.
In Madrid, normally on Oct 12 there is a large military parade attended by the King, through the centre of the city, though this year the military parade will surely be called off, so you won't have to factor that into your calculations. Of course all of this info would be irrelevant if there is a second wave of Covid-19 in the Autumn season.