I'm also having problems upgrading to the 1.8.1 versions on the Mac App store. I have version 1.7.2 of Affinity Photo & Affinity Designer, and I'm on a 2012 macbook pro running 10.12.6.
If I go to the updates tab, the App store doesn't list either of these apps as having new versions that can be updated. Things are supposed to show up there within 72 hours - it's been a week and a half.
If I go to the purchases tab, the App store recognizes that I purchased Photo and Designer, but instead of offering to let me update them it only asks if I want to open my current versions.
If I go to the Top Charts tab and find these apps, the App store lists the Apps AND gives me a button to press to Update to the latest version, but when I press that button I just get a wheel that spins forever, and no download. The same thing happens when I do this for Affinity Photo or Designer.
I'd like to download these if it's possible. Or if there's a way to transition these purchases out of the AppStore, so that I can download these directly from Affinity from now on, that would be ideal! I had no troubles at all in upgrading Affinity Publisher, which I had fortunately bought directly.