Your router probably doesn't support a VPN either then, so you would need a DNS unblocking service with support for the BBC iPlayer, something like Unlocator or one of its many competitors. The Chromecast doesn't let you change the DNS manually, so you'll need to follow the directions provided by the DNS service for how to get the right DNS addresses from your router to the Chromecast. And you'll also need to block the Google DNS addresses in your router using the static route method in order to force the Chromecast to use the DNS addresses you provide.
If you can't be bothered to do all that, there are a couple of other alternatives:
You could just use tab-casting and put up with the slightly jerky video quality that usually produces on the Chromecast.
Or you could use an app like PlayOn or Plex to relay the BBC video stream from your computer to the Chromecast, but you'd still need a VPN or DNS service other than Hola on your computer since Hola is only a browser extension. BTW, be aware that using Hola is a privacy concern because they monitor your viewing habits and sell that information commercially. There are competitors like Zenmate and Proxmate who do not do that.