How to merge two or more rasters with values 0 and 1, where the value 1 does not overlap with other rasters. Finally, I want one raster file with pixel values 1 from all raster files as 1 and the rest 0.
This should be straight-forward with a raster calculator. You could use an OR statement to generate a raster with output=1 wherever inputA=1 OR inputB=1. You could also do a simple sum (add the values of the two inputs). If the 1 values really do not overlap (both inputs never have 1 in the same pixel), then you can probably get away with just summing the two rasters, since the result of the addition should always be 0 or 1. If there is overlap, and some pixels end up with values of 2 or higher, then you can use a threshold to reduce the larger values back down to 1.
The details of how to do this depend on which software you're using (QGIS, ArcGIS, gdal, etc.).