That page you shared contains the following text:
Listeners will have difficulty accessing BBC podcasts via the Google Podcast app
So moving to Google Podcasts doesn’t seem to be a solution based on that article (?) Actually the ‘platforms’ the page talks about are things like Google Home/Assistant. But you can try of course and leave your AntennaPod as it is while trying out Google Podcasts.
This would theoretically be possible but something Google could do, not BBC. The fact that it works over 4G really points to an issue with the WiFi on your phone. The issue might be the DNS system, which translates a domain like AntennaPod.org into an IP address like 123.456.789.123 – and this could indeed be a problem that exists only on one of the two phones.