Consider this simple example (which displays in red):
echo -e "\033[31mHello World\033[0m"
watch echo -e "\033[31mHello World\033[0m"
More recent versions of
watch support color. You will need to use an extra level of quoting to preserve the quotes and escapes in the particular situation of the example in the question:
watch 'echo -e "\033[31mHello World\033[0m"'
-c, --color Interpret ANSI color sequences.
Non-printing characters are stripped from program output. Use "cat -v" as part of the command pipeline if you want to see them.
But they don't get interpreted, so I don't think there's any way.