I think Grima is originally from Rohan. I don't think he's a foreigner. I think he had been Theoden's adviser all along, or at least he must have worked within the king's court somewhere. That's how he met Eowyn and fell for her. Once Saruman learned what Grima wanted, it was easy for him to bring Grima into his service: a simple deal: Give me Edoras and I'll give you Eowyn.
Notice Eomer (or Gandalf, in the book?) asks him, "How long since Saruman bought you?" He was not always in Saruman's service. By offering Eowyn to Grima, Saruman slowly and steadily earned more of Grima's "loyalty" every time Eowyn walked by.
I think that's why Wormtongue's tear when he sees Saruman's army is so effective. He didn't think Rohan would be destroyed-- just subjugated. He begins to realize what a monster Saruman really is-- and what he has helped to set in motion that cannot, now, be undone.
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...down to the water to see the elves dance and sing upon the midsummer's eve.
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