I have not been a Jew long enough to be knowledgable to be able to answer this. However, I can see two distinct possibilities.
First, I would assume no group would allow a gentile member of the Messianic faith to use the mikveh. I do not necessarily see this as a bad thing for Messianics as it should encourage them to build their own, as well as whatever institutions they require. I do not see Messianics, especially gentiles, ever being accepted into the Jewish community and they should accept this and develope their own community and supporting instituions.
As far as Messianics who are halachly Jewish, I can see one of two things occuring.
If a Jewish messianic woman who is married to a Jew wishes to use the mikveh, a rabbi could possibly allow it because the commandment to use the mikveh is still encumbant upon her. There is also the issue of any children she bearss. I would also think this would apply to a Jewish woman who is not Messianic, but who is a member of a religion other than Judaism. A Jew who is an atheist would have no problem using the Mikveh because she has not strayed into avodah zara. But I do not know why an atheist would choose to use it anyway.
The second scenario I can see is that a Jewish woman who has accepted another religion, Messianic or not, ould be considered to have cut herself off from Judaism and from the people of Israel and would not be welcomed.
I would think that one should consult a rabbi to determine which is the case, because undoubtedly there will be varying opinions.