Do you need to be religious active to walk to Santiago or do any other traditional pilgrimage?
Most people I have heard of who have completed a large part of the Santiago route by walking or cycling were not religiously active; and even those who were practicing Roman Catholics (RC) have never made another pilgrimage.
So you will not be an exception, although the closer you get to Santiago the more likely you will meet more people doing the pilgrimage for religious reasons.
What do pilgrims do while walking?
One of the things pilgrims do (if they do a pilgrimage right) is to contemplate. Often based on questions they ask themselves or each other, sometimes based on prayer or things they see during the day.
They do not just contemplate religious things but also things happening in life. Some sample questions are:
How you are in your relation with your relatives and friends.
How you do or could help others in whatever way.
The list is much longer, I guess you get the feel of it now.
Can you do something like that when you are not religiously active?
If you want to give your pilgrimage a theme that is like them, you can do something like that yourself.
It is something you can do while walking.
You do not need to set time aside for this.
Just read one of the points in the morning or when you have started walking, think of a good point to contemplate that day, and you can spend as much or little time on it as walking, company and your mood allows.
Another traditional thing for pilgrimages is the rosary, but as non practicing RC (you say 'unequivocally atheist'), you will likely not be happy spending much time on it.
In your case I might read about it.
I was a practicing RC at the time and found the rosary boring and uninspiring.
If you are into meditation or even yoga, you might like to think about a kind of active meditation or a simple yoga routine that will get you in the same kind of mood.
This is not something for new people to meditation or yoga, it might be a good substitute in your mind.
And that is the only place where it matters.
Do I need to do something to replace religious activities?
For all I know, many people happily walk the whole distance without any thought deeper than the next meal or the next mountain to climb.
Things to do and to avoid?
If you meet people who are religious, respect it and them.
If you pass churches, you might want to go in and spend a few minutes, dress for it, (shorts till below the knee and shoulders covered, or a wrap to cover what needs to be covered.)
I am sure you know how to behave in a church as a tourist, if you happen to be in one during a service, do not take communion (or at least you should not as you have not been introduced into the church, it is your own choice of course), if you happen to walk forward with all others or if the group is small and the priest just comes to you, hold your hands in a closed position and your mouth closed.
If the priests insists, just shake your head slightly.
Most local priests will understand, they will have seen more 'non RC' people in their services but you might meet a priest out of a very religious area who is doing the pilgrimage himself.
Do donate a little if you can spend the money on it.
The people who do the Santiago route are very varied and some of them seem to feel less welcome. You can do your bit and be welcoming/friendly to all you meet.
So many churches, should I visit all?
Do not feel obliged to visit every church on your route, you will never make it to Santiago if you do.
And if one church for the whole route is enough for you, it is.
I do not want any church but I want a proper end to the route!
These days there is a route beyond Santiago to the coast, so also those who do not want to enter churches at all can have a good closure to their experience. I do not know how 'new' and how 'well known' that last bit is, I have seen it mentioned on television.
In 2018 the BBC have broadcasted a series in which several well known people did parts of the route and the full last 100 km. They did tell a lot about the route in the series. You might be able to find it on Youtube by now. Or it might be broadcasted as well where you live.