Log background job execution time

You could just use:

time yourscript.sh

But if that doesn't work for you, you can wrap your content of script like:

STARTTIME=$(date +%s)
#command block that takes time to complete...
ENDTIME=$(date +%s)
echo "It takes $($ENDTIME - $STARTTIME) seconds to complete this task..."

Just time the nohup process:

$ time nohup bash -x ~/scripts/foo.sh 
nohup: ignoring input and appending output to 'nohup.out'

real    0m10.011s
user    0m0.005s
sys 0m0.006s

foo.sh simply has sleep 10 as an example.

[this assumes that you want the output of time to also go to log.txt, something which is not clear from your question]

Just don't use nohup.

nohup does nothing special but ignoring the SIGHUP signal, redirecting stdout and stderr to a file and --only in some variants, like the nohup from GNU coreutils-- redirecting the stdin from /dev/null opened in write-only mode.

You can easily do all that from the shell:

{ trap '' HUP; time bash -x your_script args ... ; } > log.txt 2>&1 0>/dev/null &

Or safer, if started from a script or if you don't plan to ever bring back the command to the foreground with fg:

(trap '' HUP; time bash -x your_script args ... 0>/dev/null &) > log.txt 2>&1

(You can omit the 0>/dev/null if you don't care to emulate that feature of GNU nohup).

---

Of course, you can use the syntax trick from the first example with nohup too if, for some unfathomable reason, you ascribe magical properties to nohup:

{ time nohup bash -x your_script args ... ; } > log.txt 2>&1 &