Fun fact: since "szívesen" literally means "with pleasure", you can use this word if you take pleasure in doing something (so, not just as a response to "thank you"). For example, "Szívesen iszom kávét" means "I like to drink coffee", or "Péter szívesen megy iskolába minden reggel" = Peter likes to go to school every morning.
just to point out, "nincs mit" is the hungarian equivalent of "not at all" in this context but it literally translates to "there is nothing to", so that when it's said in reply to "thank you" it translates to "there is nothing to (thank)", the part in the parentheses being implied, meaning that you were doing it gladly/that it wasn't even enough effort to be grateful for. that being said, it doesn't ACTUALLY have the connotation of what you did not being much effort, it only has the same connotations of "not at all" in english, but that's just the literal translation and i thought it'd be interesting for people to know in terms of learning grammar.
in the same way, if someone were to ask you in hungarian "do you have anything to say?" you could reply with the same thing ("nincs mit") and it would mean "there is nothing to (say)"