HSTS tells web browsers that your non-HTTPS site must never be used. If this was set at the old host, then web browsers (probably yours, but also lots of your visitors) will always try to visit https://example.com not http://example.com regardless of what they type, or what bookmarks they follow. It sounds from your description as if HSTS is set at the old site, although it’s hard to be exactly sure without a bit more technical detail because it could also just be redirecting you “by hand” without HSTS.
Let’s Encrypt will be happy to issue you with a new certificate at any time, no need to wait for the old one to expire (but there are rate limits, so please don’t try to do this every day). It makes sense to ask for one as soon as the new hosting is working. However, if you have HSTS then the unencrypted HTTP site won’t work at all, so it will be especially important to get the HTTPS site working ASAP. Without a valid certificate (and associated private key) the HTTPS site won’t work and there will indeed be a warning. In fact, if HSTS is set, it’s not a warning, it’s a fatal error, many browsers just won’t let the visitor see your site at all until the problem is fixed.
So: If it’s practical get the valid certificate and private key from the old site, you should do this, at least as an interim measure. Otherwise, make sure before you do the transfer that you understand how to get a new Let’s Encrypt certificate at the new location and have it installed for HTTPS, and put both very high on your TODO list once the new host is serving your site instead of the old one. If HSTS is set, and you find you can’t get a valid certificate installed for the new site, you’re going to be “off the air” until you fix that.
Also if you’re a blogger this is a good time for a “heads up” post saying there’s going to be roadworks and for people to please be patient / reach out via some other means until it’s all finished. Happiness is the correct setting of expectations.