I'm not sure what problem people have from the ISP by using a VPN; Comcast even has a page on their web site about using a VPN:
Comcast and VPN use

The only problem I can think of is that the connection bounces through a number of routers in various locations. Since it doesn't take the most direct route, the speed may be slowed down a bit, and that could affect how well the program streamed. A lot of VPNs are cheap; you could sign up with one for a month and see how it works. Or you could try this:
Watch BBC with DNS service

I, personally, have not used a DNS service, but it looks like it is an easy way to go.

I have experience with these techniques going in the opposite direction. To get one question out of the way, VPN encryption is very commonly used in the US (and everywhere else) for confidential connections to one's employer's computer systems and other purposes. It's normal traffic, your ISP won't care.

"Broadcasters" and other content providers (like Netflix) have legal and licensing reasons why there are geographic restrictions on use. As an example, were Netflix to license Star Wars for US customers, it doesn't have the right to serve it up to a user in Japan or Jordan, but someone in one of those countries could use a masking service to connect as a US customer. For licensing movies and TV shows, rights for each location must be purchased separately. As I understand it, the BBC doesn't have the legal right to provide its normal domestic programming outside the UK because domestic users pay an annual tax for the content.

With that background, there are websites that will disguise a user's actual location by one of two technical means, a proxy server that relays the connection and appears to be in the country of signal origin (kudzu called it DNS) or a VPN connection (an encrypted connection that similarly emerges in the country of content origin). These connections are not that hard to detect because the content providing site will see numerous connections coming from a similar range of IP numbers that are registered to companies that aren't ISPs or other telecommunications carriers.

As far as I know, the legality of using a masking service isn't a real issue (to the user) but they sometimes get shut down. Netflix announced recently, probably as a result of pressure from its content providers, that it's considering whether it will do something about this practice. I'm sure the same is true for the BBC, there are many British expats using these sites to watch their favorite shows.

A typical charge is $5-10 per month for a geo-masking service.

We watch BBC on a Samsung Smart TV. I had to download the free app on SmartHub. I do not watch news on a computer.

Brushworks-

I think the issue is that there are many BBC shows that are only available in the UK, and not on BBC America.

Yes we have to be station specific. There is even a BBC World News shown in America.

For me, I've tried streaming CBC to no avail.

Thanks for all the comments. I think I'll try it and see what the services are like.

What shows are of interest? There's a ton of them on Roku. The Filmon channel (a private channel) offers over 600 free live tv channels.

AcornTV is British, Canadian and Australian TV. $5 a month on Roku or some Smart TV's.

Acorn TV

If you want it, and take time to look for it. it's available on the internet.

The shows on Acorn of BBC origin are old, not current. Imagine someone had just moved abroad, wanted to watch US TV, and someone pointed you to a website that was showing Ed Sullivan Show reruns. Not quite that bad, but you get the idea.

There are no free or cheap sources (of TV content or software, for that matter) that have geographical restrictions and/or that some have to pay for.