the booking and attendance plugins don't "talk" to each other - but if you are upgrading your Moodle site you will need to upgrade any 3rd party plugins you have installed on your site as well - you can get updated versions of most Moodle plugins from the plugins database here:
Alternatively you might try to convince your department that it's not something you are experienced with and in the longer-term it might be safer/cheaper/more reliable if you get an experienced Moodle partner to handle this for you.
If you have a smaller organisation you might take a look at the MoodleCloud offerings provided directly by Moodle HQ:
https://moodlecloud.com
Or alternatively there are a number of Partners in the UK that you might want to talk to: https://moodle.com/partners/?country=GB
(FYI - I work for one of the partners in that list)