I run the client currently on 2.0.61 BETA (March 30th 2023). The client offers me an update to version... 2.0.61

When trying to update, I get the known update error message. And when I try to manually look for updates, the process just runs forever.

And even though I usually got the update error messages since 2.0.58, the client updated anyway.

So, in which future version will this issue be fixed?

I have experimental updates enabled in version 2.0.61.63 and GOG Galaxy doesn't offer any update. If I check for updates manually, I get the message that there aren't any, as expected.

If you want to report a bug, open a support ticket and attach your zipped up C:\ProgramData\GOG.com\Galaxy\logs folder. Otherwise, you're just hoping they're going to reproduce the issue themselves. That's unlikely since it's not a universal problem (like the broken autoupdate they fixed recently).

If you want to try to fix this yourself, perform a clean install for the non-existent updates being offered, and double-check your firewall and make sure IPv6 is disabled for the manual update check never completing. I can post directions if you want them.

This issue was fixed weeks ago. I suggest downloading the Galaxy installer from the website and installing it again on top of your current installation. That should let you update to future versions.

The updater itself was broken, so it can't install new versions on it's own. That's why you need to do it manually.
Try downloading the current version and install it over the current installation. But do so right after Windows start, before Galaxy was launched.