The PuTTY website has this marked as a request for enhancement made back in 2017. Does not appear anything has been done.
However, there is a slightly messy way to make this happen. First, setup PuTTY to log to a file, (example uses /path/to/log/putty/log). Make sure PuTTY is setup to with "Flush log file frequently" checked in the logging options.
In a bash terminal window, use tail to follow the log file and pipe that to ts (timestamp). Something like this:
tail -f /path/to/log/putty/log &> >( ts '[%Y-%m-%d %H:%M:%S] ' ) \
| tee /path/to/timestamped/log
Using tee is optional if you don't care to see the output. Output is something like this:
[2023-08-01 21:29:19] user@pi:~ $ uname -a
Linux web-pi 5.10.103-v8+ #1529 SMP PREEMPT Tue Mar 8 12:26:46 GMT 2022 aarch64 GNU/Linux
[2023-08-01 21:29:31] user@pi:~ $ lsb_release -a
No LSB modules are available.
[2023-08-01 21:29:31] Distributor ID:Raspbian
[2023-08-01 21:29:31] Description:Raspbian GNU/Linux 11 (bullseye)
[2023-08-01 21:29:31] Release:11
[2023-08-01 21:29:31] Codename:bullseye
putty timestamp