In the datasheet that you provided are links to more reference documents. One of them is ATWINC1500 Wi-Fi Network Controller Software Design Guide. In chapter 13 is described the WINC Serial Flash Memory, including how to access it.
In case that this doesnt work you can also disassemble ATWINC-1510 module and find direct access to the flash IC and read ...
binwalk scans files for magic bytes. Data that does not have a signature defined for it will not be found by a signature scan.
It appears no signature is defined for RootFS file systems (the hits for rootfs below are labels for offsets in TRX headers):
Example entries for SquashFS file systems signatures, for reference: