It seems like every single British cop show I’ve ever seen, at some point has the family of the victim going on TV to issue a statement and plead for the return of their loved one. Does this happen in real life? I guess there probably aren’t (relatively) that many cases of people going missing but when someone does, is this a common thing? I don’t think I’ve ever seen it happen in the U.S. But maybe I’ve just missed it.
John Walsh launched a career out of doing this.
I’ve only seen a few episodes of America’s Most Wanted so didn’t know he did that kind of thing. The British thing seems to be people just hopping on live TV to do their appeal. Not on any particular program. They’re usually flanked by police when they do it.
Which stemmed from him trying to find his son, then his son’s killer. Now he has an entire TV channel which features him or his other sons spotlighting missing children during at least one commercial break per hour.
Are the families getting up on stage with the police though?
You didn’t mention that in the OP. 
So, does anyone know if this is actually done in Gr. Britain like I’ve described?
It’s done sometimes, particularly for missing children. It’s done alongside the investigating police officers, at a press conference, and it’s more aimed at being on the national (or regional I suppose) network news programs than on live TV as such. It’s mostly to raise awareness, but it’s not unknown for the suspected perpetrator being the one doing the appeal.
Yes. You can find lots of videos of such press conferences online.
Does this make any difference? I’ve no idea. It may well be that their main use is to help the family feel that they’re doing something practical to assist the investigation.
Susan Smith did this after she killed her own children.
In Silence of the Lambs, it was a “smart” thing to do, since it supposedly humanized the victim, making the perpetrator less likely to treat her as an object.
It’s reasonably common and tends to be shown on news programmes.
The other police PR trope we get is when a senior investigating officer is put up to sermonise after sentence in some particularly newsworthy case. Always makes me cringe - IMO that’s the judge’s job. Yes, the police did well to get the evidence and put the case together, but that’s as far as credit should go. They’re not moral arbiters and shouldn’t pronounce as though they are.
The police and the prosecutor are the advocates for the victims. The judge is supposed to be neutral and make sure everything is done legally.