As @Seth said in the comment, your ISP is the one that is blocking your access to these websites. What you describe seems perfectly normal depending on the rules your ISP has established.
You mention that you can access the websites with the second ISP, and I am assuming that the same applies with your mobile data.
I had the same problem with SKY in the past. Some ISPs have filters embedded on the routers or through their DNS that prevent you from accessing websites (or for parental control). The way these filters work is really confusing, I remember accessing sites (subtitles) that should be blocked (according to the strict rules of UK) instead they blocked me from accessing sports news websites from outside the UK commenting it as file sharing website.
Now every time you called your ISP for this issue, they might have whitelisted this website, for a certain amount of time, but when they update the filters it will be blocked again. Also, these filters are not always on, there are periods of time during the day that are automatically disabled (usually at night). You can always disable these filters but this will not guarantee you that your ISP might not continue blocking you. Keep in mind that filters do not always use the best rules for blocking websites, so blocking you from accessing these websites might seem strange to you, but perfectly normal to the ISP.
My solution to these issues was to use a VPN. Every time I want to do something on the Internet (read,browse,watch tv) I use VPN. There is nothing wrong in using VPNs (for legal purposes), on the contrary it adds a layer of privacy and security. VPNs can give you access to blocked websites, geo-restrictions, and most importantly:
It secures your computer's internet connection to guarantee that all of the data you're sending and receiving is encrypted and secured from prying eyes.
Source: Why You Should Start Using a VPN
I wouldn't worry so much, and get frustrated about getting blocked since there is an easy and common way to fix that. And as you already said you have used VPN and worked like a charm.
This answer proposes the use of VPN for legal purposes only.