Emulating a do-while loop in Bash

Two simple solutions:

1. Execute your code once before the while loop

   actions() {
      check_if_file_present
      # Do other stuff
   }
   
   actions #1st execution
   while [ current_time <= $cutoff ]; do
      actions # Loop execution
   done
   

2. Or:

   while : ; do
       actions
       [[ current_time <= $cutoff ]] || break
   done
   

Place the body of your loop after the while and before the test. The actual body of the while loop should be a no-op.

while 
    check_if_file_present
    #do other stuff
    (( current_time <= cutoff ))
do
    :
done

Instead of the colon, you can use continue if you find that more readable. You can also insert a command that will only run between iterations (not before first or after last), such as echo "Retrying in five seconds"; sleep 5. Or print delimiters between values:

i=1; while printf '%d' "$((i++))"; (( i <= 4)); do printf ','; done; printf '\n'

I changed the test to use double parentheses since you appear to be comparing integers. Inside double square brackets, comparison operators such as <= are lexical and will give the wrong result when comparing 2 and 10, for example. Those operators don't work inside single square brackets.