I assume that is the start time, so you can see the the part ... END TIME =`date +"%r"` was executed at the time of invocation of the ssh command. I think you need to split it into 2 parts:
Code:
....
START=$(date +"r")
address="/home/campus27/zwang10/Desktop/AWRR/program/power/vmodel_1/nprot/K_10"
ssh -f "${host_list[0]}" "cd '$address' && nohup Rscript '${program[0]}.R' > '${program[0]}_sh.txt'"
END=$(date +"r")
echo "The job\n $address\n${program[0]} is finished\nSTART TIME = $START\n' END TIME = $END" | mutt zwang10@mtu.edu -s "${host_list[0]} - Job ${program[0]}.R finished" -a "$address/${program[0]}_sh.txt"
(not tested)
It will execute the echo and mutt on the local host, and your original example executed them on the remote host (if that counts).