Ideally you would have a changing table with a mat that has a cover which can be wiped after changing each child. All the better if that can be in a room where there`s a sink for handwashing but not all washrooms have that much space. Just outside would also work. There`s no reason to be scared since you are never leaving a child unattended during a diaper change. What scares you about it?
Worst case, I guess you could us the mat on the floor but check your provincial requirements first. I know some provinces require the changing area to be in a room with a sink dedicated to handwashing.
The main things to consider are that you should be washing your hands before and after changing each child. That the surface of the mat must be wiped after changing each child. That you shouldn`t be walking through the house or day care to reach a sink each time a child is changed due to the risk of touching something on route with fecal matter on your hands. Most childhood illnesses are spread by body fluids (coughing, sneezing, fecal matter) which is why hand washing is key to ensure spread of illness is kept to an absolute minimum.
I don`t think parents will be overly worried if you don`t have a change table due to space restrictions. But being able to show them that the surface their child is changed on is easily disinfected between changes, and that you have easy access to wash your hands are really important.