You need combination of SDBlockDevice and FATFileSystem with File.
Currently I have no actual example for that but I think this old one is useful.
SDCard - Example of using SD card with Mbed OS5
Just try it in this state, and later try to update it to latest Mbed OS. I think it will working.
Anyway the example from FATFileSystem contains all necessary things, I think.
You can replace the default BlockDevice with the constructor of SDBlockDevice, uncomment FATFileSystem and commented-out the LittleFileSystem.
Let me know
BR, Jan