I have to say good for you for being the bigger person and not telling the kids to call you that anyways, or consipiring to have them call you that when they are around you. That would be tough, I imagine. I agree with you that the mom is threatened by you, and the whole situation is a sticky one. It isn't fair to you, but also keep in mind that it is hard for bio mom to have this competition for her children's affection.
Ultimately, the kids are what is important. And not putting them in a difficult position is what will win you this battle in the end, even if they never call you Re-Re again. I think you are probably right in feeling like there is a little bit of distance that wasn't there before. But I think it is probably because they are nervous that you will get angry with them for following mom's instructions. Because of course you are going to notice that they call you something different, and they are waiting for the inevitable question of why. By not making an issue of it, you are showing them that they are what is most important, and that you love them, and you are not going to make them pick sides. I think the bond will come back when they relax a little and realize you aren't mad at them and that you care for them as much as you ever did. IF it comes up again, I would just say that they should call you whatever they are comfortable calling you, that your name doesn't change your love for each other. You could make a joke and say something like "as long as you dont start calling me fish-face, that would be a deal breaker" or something else that would make them laugh. Lighten the mood, and let them know it isn't an issue.
Hopefully this is as bad as it gets, and you will all be just as strongly bonded as ever. It isn't a completely bad thing to have a little bit of maturity and authority in their eyes. Maybe she did you a favor as you head into the teen years and she doesn't even realize it. Good luck to you all!