Hello,
as deexno has instructed me - I have done the troubleshooting by using:
systemctl stop checkmk-telegram-plus-<omd_site_name> && /usr/sbin/runuser -l <omd_site_name> -c 'cd /omd/sites/<omd_site_name>/local/share/checkmk-telegram-plus/ && /omd/sites/<omd_site_name>/bin/python3 /omd/sites/<omd_site_name>/local/share/checkmk-telegram-plus/telegram_bot.py'
which revealed that the old API KEY has remained after the update.
File "/opt/omd/sites/chekmkfree1/local/share/checkmk-telegram-plus/telegram/_bot.py", line 546, in initialize
raise InvalidToken(f"The token `{self._token}` was rejected by the server.") from exc
telegram.error.InvalidToken: The token `5917540767:nnnkEaZ5cpwttTrtndL_MKkT8AKwODIbcAA` was rejected by the server.
Hence I have uninstalled the plugin and rm -r /omd/sites/<omd_site_name>/local/share/checkmk-telegram-plus, which enabled me to do the successfull “first install”.
From there all went smooth and am now running three separate bots for as many sites 
However - I have noticed that for the use of a bot to query the CMK - the notifications does not have to be configured and (for me) it does not work if configured (no notifications to the bot is delivered automaticaly).