They’re talking about a number that appears on the face of a clock radio (not a spoken time given on the radio)

Фраза странная, особенно в сочетании со вторыми предложением. Я вообще не вижу смысла спрашивать сначала время, а потом не опаздывают ли они НА САМОМ ДЕЛЕ. И почему именно clock radio, а не просто любые часы?
И встречается только в русских учебниках. Не удивлюсь, если имелось в виду прибытие самолета и объявления о прибытии в аэропорту.

It’s not at all strange to me. It’s a number that appears on some sort of radio, be it a clock radio or a radio interface on a plane, but it’s not an audible announcement of the time over the radio.

REALLY here doesn’t mean ДЕЙСТВИТЕЛЬНО, it means ОЧЕНЬ/СИЛЬНО.

Удаляю. Если хочешь, добавь сам.
Часы-радио / часы с радио / радиобудильник

You add it - I wouldn’t know what to say. Just bear in mind that they’re LOOKING AT (not HEARING) the time on some sort of machine with a display to see if the people they’re waiting for are VERY late.

Спасибо, но не буду. Я понимаю REALLY, но в радио с часами смысла не вижу. Пусть Елена сама переводит - она хоть контекст знает.

If you look at this logically, the radio doesn’t constantly announce the time, so they’re talking about something that can be checked at that moment to verify the time by LOOKING at it.

I do look logically - and it makes no sense to me to say 'radio' even if it's the only device around showing the time.
Possibly, you are right, but to me this looks like some very special context - and google doesn't find anything.

You don’t need google for this. When we talk about a visible time reading, be it on a watch, a clock or an analog device, we say THE CLOCK SAYS... Similarly, here, the radio SAYS because they’re looking at it.

https://www.google.com/search?sxsrf=ALeKk03XbmMa5Dwov0fahvWYf_PqIou6uA%3A1615409525921&ei=dTFJYIbmN8y7tAbq4qugBw&q=%22the+clock+says%22&oq=%22the+clock+says%22&gs_lcp=Cgdnd3Mtd2l6EAMyAggAMgIIADICCC4yAggAMgYIABAWEB4yBggAEBYQHjIGCAAQFhAeMgYIABAWEB4yBggAEBYQHjIGCAAQFhAeOgcIABBHELADOgQIIxAnOgQILhAnOgUILhCRAjoNCC4QsQMQxwEQowIQQzoECC4QQzoKCAAQsQMQgwEQQzoLCC4QsQMQxwEQowI6CAgAELEDEIMBOgUIABCRAjoLCC4QxwEQrwEQkQI6CAguELEDEIMBOgUIABCxAzoECAAQQzoFCC4QsQM6CAguEMcBEK8BOgcILhBDEJMCOgUIABDJAzoLCC4QsQMQyQMQkwI6BwgAEIcCEBQ6CwguEMcBEK8BEJMCOgcILhCHAhAUUI-5JVi9yyVgiNAlaANwAngAgAGlAYgBrhGSAQQwLjE3mAEAoAEBqgEHZ3dzLXdpesgBCMABAQ&sclient=gws-wiz&ved=0ahUKEwjG9NW3zabvAhXMHc0KHWrxCnQQ4dUDCA0&uact=5

Yes, 'clock says' is fine. My problem is with 'radio'. Even if the only clock around is on/with radio, personally, I'd always say 'clock', not 'radio'. To take this to extreme almost every appliance has a clock nowadays - microwaves, ovens, coffee makers, smart thermostats ...

And they all SAY the time every time you look at them 😅

Yes, but if I need time to know if somebody is late or not, I don't care where the clock is and wouldn't ask specifically about the clock on the microwave.

Ok, let’s say you’re in the bedroom and I’m in the living room. I don’t know the time - I’m in the living room waiting for someone to arrive. I know they were supposed to come at 4 and I know they’re late, but since I can’t see the clock radio, I have no idea HOW late they are. So I say “Grumbler (take a look and tell me) what time does the radio say? Are they REALLY late? (...or just a little late?)

OK, if you insist, but I'd never mention radio even in this situation.

My guess is that one person is sitting in a car and one person is outside at a station waiting. So they mean the car radio, which always displays the time.

Ну, тогда просто радио