Hawa93, I don't think you're BU. As you say, if DD began calling MIL "Mama" organically, b/c she heard the other grands calling her that, then it would be ok. But I can see where the fact that MIL seems to be pushing it would be disturbing.
I agree w/ others, however, that if DD comes to call MIL "Mama,' she will know the difference between that and her actual mum (you). We adults know that Mama actually means "mother," but to DD and her cousins, it will mean "grandmother" for a long, long time. Even when they become conscious of the fact that usually it means "mother," they will know in their family, it refers to MIL/this particular GM.
Maybe MIL became used to being "Mama" b/c of SIL's kids and it has become part of her "identity." But the fact that she now is trying to blur the lines between Mama and Mummy - and between DD and GD - has me concerned. It DOES sound as if she's trying to "play mummy" or, at least, be thought of on the same level as mummy/as a 3rd parent.
It also bothers me that she continues to insist on Mama after you and DH told her how you two feel. And it bothers me that she, apparently, complained to SIL about it, and that one or both of them decided it was all coming from you, even though DH is the one who told her to "quit it."
IMO, you and DH need to talk this over and decide what your priority is - to keep MIL from teaching DD that she's Mama or keep her from mixing up the terms mama and mummy, etc. (the second is more important, IMO, but that's just me). Then, you'll have to approach her as a united front and let her know exactly what your boundary is. And you'll have to enforce it every time. This can be done gently-but-firmly, as in, "No, DD, this is Nana, not Mama" or "No, in this house Mama and Mummy are NOT the same" (caps for emphasis, not yelling).
I know it seems like a petty issue to some people. But, IMO, it's the sign of some larger concerns (MIL trying to be in control, disrespect for the parents' feelings).
If you decide that you're ok w/ Mama as long as she doesn't interchange it w/ Mummy, then you may want to let SIL know about that incident. That may concern her, too, and she may want to be on board w/ you in making sure that doesn't continue.
If this continues to be a problem, you may want to cut back your visits w/ MIL. Maybe you do, anyhow. I'm NOT saying to cut her out altogether over this. Not at all! Just saying that the less DD is around her, the less likely she is to be influenced by her. Meanwhile, you can remind DD of who MIL is or what you feel she should be called and just say she's "being silly" when she calls herself "Mummy" or whatever. It's very easy for parents to do this, especially if you're with them a lot more than the GM. (As a GM, myself, I'm not crazy about the idea, but some GMs force the issue). DD will believe YOU.