I think people in the UK are unlikely to encounter someone who will wish them to "have a blessed day". If it happened to me I would probably just smile and walk on. Of course, some people might use it as an occasion to vent their frustrations and third degree interrogate the well wisher as to his or her intentions.

"Yeah. Right." (backing away slowly)

It's words, generally meant to convey good feelings, .....I say thank you.......all smiles, hugs, and good feelings are welcome.

Depends on who asserts that brown sticky nugget of nonsense.

If they are Not in a position to potentially make my life miserable and are attempting to be snarky, I'll respond with "Blessed?" with exaggerated hand, body, and facial expression to clearly express "WTF" then adjust hands (open hands pointing at them and facial (dumb idiot) jester to express: "Just how stupid are you?" finishing with eye rolls, sad shaking head in disgust.

If they are a potentially trying to be nice, I'll just roll my eyes and reply with a slow "O K? well, you have a nice day then." People that know me have long ago learned "Have a blessed day" is not a beneficial statement.

I don't.

How would you respond to something like "may Thor bless you"? How is it different?

And you as well!

The first time I heard it, I thought it was some sort of pagan thing.
So you could respond with
Merry meet
And merry greet
And merry meet again!

I was in a co-op the other day and two employees stocking shelves were talking about how to answer when someone ask how are you doing. I interrupted and said sometimes I answer 'terrible'. Some just nod and go on (because they're not paying attention) but some say OK and then hesitate and say 'what.' That always get a conversation going. They liked this and said they would try it.

"HAve a Blessed Day." is the standard cap to just about any conversation, long or short, in the area where I live. I find it as irritating as being told "Have fun in hell," which is what I know they'd be saying if they knew I was an atheist. My response is usually just a tight-lipped non-smile as I walk away. I know if I actually used words, it wouldn't be pretty.