That's a big question. First, I think society will have to entirely rethink what it means to earn a living. If the robots take as many of our jobs as it now appears - and there seems to be a machine to do just about everything it seems. Will it be leisure time or just suffering and want? That dynamic, I think, will have a great deal to do with how the religious will react to it. If it is suffering and want, I could see religion getting a boost as people flock to it for comfort. On the other hand, if it means more leisure, I could see a lesser increase, as people delve deeper into their interests and passions to give their lives the meaning once provided by work.

The world has undergone similar things in the past; look at the social upheaval at the start of the industrial revolution for the answers to your question. People will be displaced poverty will become rampant new social norms will occur. More religion or less there will be a shift in beliefs, it is likely to generate lateral shifts in religious belief more than an increase or decrease in the amount of deluded.

Perhaps if/when they begin to understand the validity of scientific application and reasoning, insofar as it has and will produce new and better ways to take the drudgery out of people's lives; perhaps then they'll begin to realize that the mythology of deism is no longer necessary. Or, maybe it will give them enough time to conceive of even crazier myths.

Believers certainly won’t entertain the idea that AI has consciousness. The trans-humanism crowd already has a religion centered around AI. Hard to say how we’ll react when faced with super-intelligence far greater than our own.

I'd like to think we will use the coming wave of technology to usher in a new wave of leisure and prosperity.. however, it doesn't seem to have been utilized that way up to this point! To date, we seem to utilize the ability to do more with less.... to make more profit! You'll have a hard time convincing capitalists, that they should "slow down", now that we have robots. For example... say you are a ditch digger... and someone gives you a magic shovel that let's you dig the ditch twice as fast with the same amount of effort... now, you get paid by the job... You could stop in the middle of the day and take the rest of the day off... OR, you can go dig another ditch and take home double the money at the end of the day. Seems like option 2 is winning. I'm not so sure we are going to get that leisure with Capitalism in the driver's seat. Anything but a model of "growth" is totally unacceptable to the capitalist ethos...but perpetual growth simply isn't tenable, so something has to give. Like it or not, we are gonna need a little socialism.

I don't tend to think religious adherants will have their faith influenced much one way or another by technology specifically. I think one thing religion provides that secular life may be missing is rituals of embodied connection. Until there is some similar secular method to fill what seems to be a necessary function to cohere a society, religion will persist. People need more than an academic knowledge of who they are in relationship to society... i think we want to feel in rapport with it... and that is something a shared practice helps to engage.

I don't think anything will change

First, look long and hard at what a believer really is.

It's a pretty sure bet that they'll just say robots are the work of satan, if they're affected negatively.
If they think robots have improved their lives, they'll give the credit to their god.

I don’t believe having more time to think will have much effect.

They'll hate them. "They're taking our jobs".

They'll think the devil is controlling the machines.

Technology does not affect their beliefs or it would have done so by now. Big social changes generally happen after a generation or two die.