How to update c5 if site doesn't know there is an update available?

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I'm running c5 4.4.1.1.
It is not connected to the community due to a site token error, so
my site does not 'know' there are updates available. So how do I update manually or rather, I would just like to point out to my site that there is an update downloaded ready for installation.

I tried putting the zip file of the update into the files/tmp folder where I thought concrete might download to but I guess I have to do something to tell concrete an update is available.

Would it work if simply replaced all the core folders with the core folders of the updated version maybe? It seems rather crude to me, and would I not run into any database issues that way?

rritz