Some of the external disks on the market come formatted in NTFS with a utility to allow macOS to write on them. (Natively, macOS can read NTFS volumes but not write on them.) The reason I inquired about the formatting of yours was to see if you had lost that NTFS-writing program when you upgraded to Catalina, which seemed like the most likely explanation, as Bob suggested.
If you have since verified with Disk Utility that your volume is truly FAT or EXFAT, the above would not apply. (If not, please do so.)
I have not yet tried using a FAT or EXFAT volume with Catalina, so I cannot report any personal experience. Apple's online documentation states that both FAT and EXFAT volumes are supported in Disk Utility.
There were reports in this forum of EXFAT volumes suddenly going read-only in previous versions of macOS. At least one was corrected by resetting NVRAM. (See Reset NVRAM or PRAM on your Mac - Apple Support.) I regret that I have no assurance this will help in your situation.