The BBC is legally obliged to block overseas access to its services (because they don't hold the international broadcasting rights to most programmes).
So, obviously, they block access from overseas IP addresses. Also, since more and more people are now using VPNs, they now also try to work out which 'UK' addresses are actually VPNs (either simply by analysing the volume of traffic from them or by subscribing to VPN services themselves and then seeing what IP address they've been allocated).
So with anyVPN (whether it be TunnelBear or anything else), there's an increasing risk that you'll find that it won't work to get access to BBC content from overseas. That's not to say that no VPNs will ever work. (My friend in Italy is still able to use a VPN to watch BBC content). However there can be no guarantee that any particular VPN will continue to work, as it's always a constant cat-&-mouse game between VPN providers and content providers.