A few weeks back I asked what to do if the men’s mikvah has a Shem HaShem on the wall… it was suggested to somehow change that, if this is not a possibility: Should I wear a towel and then turn around so only my back-side is facing the Shem Elo-heinu? Is it better not to go to the mikvah at all?
If the Shem Hashem is written on the wall, and going to the mikvah means one will have to expose himself before it, then it is better not to go to the mikvah at all.
Note that this applies to a Shem that is written as the Name is written in the Torah. For Elo-heinu, there is no vav, and no hei.
The Shem should be covered up with a poster or some other method, and then there will be no problem of visiting the mikvah.
Please see here for further details.