Yes, you should absolutely wait and not combine two drugs that suppress brain function, like diazepam and codeine. Your concern about mixing "downers" was totally warranted.
Your paramedic or EMT that responded was right, but smart-ass sarcastic. He's probably used to non-breathing blue-lipped fent-laced heroin ODs, but you have combined essentially the recommended dosage for each drug. Remember, that guy goes to sleep at night with the faces of dead mangled car-wreck victims and open-mouthed dead-eyed relatives as he tells them their relative is gone, stuck in his mind forever. So I'll forgive him for exaggerating your safety.
The catch is that diazepam, though it lasts a while, is not particularly harsh on your breathing in overdose. It is a central nervous system depressant, and that will eventually stop your breathing, but it takes a massive OD on benzodiazepines to do that.
30mg of codeine is just the recommended dose for pain in an adult. It's not particularly potent, and although your paramedic forgot that there can be complications from codeine in fast metabolizers, even if you were one, you just didn't take enough.
So for future reference, it's totally not something to worry about. Six hours between downers of recommended dose, in this case, is plenty of time.