In general, yes, you can reset the needle bar to accept a standard 15x1 needle. However, there are some quirks. Two that I've run into lately are a pinned needle bar and a closed bobbin case.
Most needle bars are held in place by a set screw-these are easy to move. Some have a screw pin that goes all the way through the needle bar. Occasionally, these will also have a set screw and you can just use that and skip the pin altogether. Oh-and sometimes the hooks are pinned in place as well, making retiming them a colossal PITA. It makes sense to the original machine; if both the needle bar and the hook are pinned in place, you can't ever have timing issues. But it makes it anywhere from a challenge to impossible to change things later down the road.
About bobbin cases. If the bobbin case has an open 'notch' where the needle comes down into the bobbin thread, make sure that you changing the upper geometry isn't going to interfere with the lower geometry. This can sometimes be corrected by cutting a notch in the bobbin case (I've seen this workaround for the Singer 319). My understanding is that this is mostly an issue when the original needle has a substantially longer bit between the eye and the point than a 15x1 does.
I'm fairly sure there's something I'm forgetting about needles, but someone who knows more than I do will chime in, I'm sure. Good luck with whatever you decide to do!