It takes 25 hours for an egg to process through the hen's internal egg-laying factory. The first 15 minutes of that process is when the egg can get fertilized. The rest of the 24 hours 45 minutes, the egg cannot be fertilized.
Let's walk through it. If the rooster mates with the hen on Monday, Monday's egg will certainly not be fertile. Odds are pretty good that Tuesday's egg has already started through the process and is beyond the 15 minute window for fertilization, so don't count on Tuesday's egg being fertile. It's possible but really doubtful. Wednesday's egg will probably be fertile.
I've read different opinions, but most agree that a hen will probably stay fertile for 2 weeks after mating. In your situation, I'd use 10 days to be safe. With 10 hens, you should get all the eggs you need to incubate with that 10 day window.
Notice that this is after mating. Just because a rooster is in a pen with hens does not mean that all are automatically mated. With 10 hens, it is very possible he would not mate with all 10 the first day. It does depend on the age of the rooster and the hens, the vitality of the rooster, and his personality. If the rooster is immature compared to the hens, they may not let him mate with them. He has to be able to establish dominance over them. When you put them together, you should be able to tell within a few minutes if this is a problem. An older rooster will possibly not have the vitality and urge that a younger rooster would have. And some are just more aggresive than others.
I understand and applaud your wish to not push your luck with your neighbors. It shows responsibility on your part.
Another thing to consider. If you bring in a rooster from an outside flock, you take the chance of bringing disease or parasites into your flock. With your scenario, you cannot do the 30 days recommended quarantine. How much of a chance depends on how isolated and closed the other flock is. I'm not saying you are guaranteed a problem if you do this, just that you need to recognize there might be a problem.
Have you considered just getting some fertile eggs from your friend, maybe trade eggs? It would make life a lot simpler for you and reduce the chance of introducing disease or parasites.