Will this significantly change the omnidirectional nature of this antenna?
Yes. For 270 degrees of azimuth, you have no ground plane. Or another way of looking at it: you are missing 3/4ths of your radials.
There will still be some radiation in every direction, but you may find that there is significantly less radiation in the direction where you have no ground plane, and the take-off angle in that direction will be higher as well.
The picture is additionally complicated by the fact that you do have a sort of "ground plane" even where there is no roof: the Earth. There will be significant capacitance between the vertical element and the Earth, meaning Earth is going to pick up some of the ground plane's job. The Earth is very lossy, so this can't do anything good for antenna efficiency, and Earth is also not at the base of the antenna, so the feedpoint impedance will probably not be what it would otherwise be.
Depending on how you route the feedline, that may appear to the antenna as a radial (there being no better options around), and you may have to do something about the resulting common-mode currents.
It's hard to quantify exactly what the result will be, since it will depend on your particular geometry and Earth conditions. If you have access to some modelling software you could find out, but in any case it's safe to say the result will be less than ideal. It will work, however, as any number of sub-optimally mounted mobile antennas demonstrate.