We just found the cause. It was a plugin called “Impressum manager” – it had a META NAME=”ROBOTS” CONTENT=”NOINDEX, NOFOLLOW” in the source code. We deactivated the plugin – problem solved.
You can only find the tags by literally looking in the source code of the plugin: Plugins => Plugin editor > Select plugin (top right) – click in the code and use STRG F to search e.g. for “noindex”. We literally went through each and every plugin. Of course there’s an easier way – to run a script though the .php files on the server – but if you have no access to the server this is the manual way.
In the meantime we also discovered another plugin called “flowpaper”, which had the same META NAME=”ROBOTS” CONTENT=”NOINDEX, NOFOLLOW” tag in the code. We also deavtivated this one too.
I thought I’d post the solution incase other people are having the same problems. It’s unfathomable to me why a plugin would include a tag like this that has huge ramifications on your entire website!