Do any of the templates (running on website since 2015) need to be uploaded again to be compatible with joomla 3.7?
Sorry, meant eLearning template.
Template from 2015 is too old version, and using Helix II framework, not Helix III.
So, you should setup new LT eLearning, renew license to get license for your website, set up it on DEV domain. Then, you bring your current content to new DEV site with J3.7, after finish you can replace your DEV site to your live website.