I do not have ILR and I have a mortgage! It can be done.
We worked with a mortgage broker who ensured she did not show us any options that would not lend to us. In the end, we only had a 10% deposit (we kept our other home and converted it to a Buy to Let, but that house is only in my husband's name. It had all of our equity tied up in it). We had a pretty competitive rate on a 2 year fix term.
The one thing they DO want, is for you to have a decent amount of time remaining on your current visa. That's why we applied straight after I was granted my second FLR(M). If I had had less than a year or 18 months on my visa, it would have been more restrictive.
Good luck!