A native speaker is likely to say something like this:
Because of roadworks I will not have internet access until (some time) next week.
It would NOT be idiomatic to use until after the present continuous: as in: I am not having (internet access) until.....
and we generally do NOT use the present continuous in the context of NOT HAVING SOMETHING except with expressions about abstract concepts such as: I'm not having a good time or I'm not having much luck.
We wouldn't say I'm not having any pets or I'm not having any siblings. Here we prefer I don't have.
Other options are:
I won't have / I will be without ....internet access until...
In short, the present continuous tense does not work in the context you describe.