If a read is in progress when the 10 seconds times out, then the heartbeat will be delayed until the read completes.

Sure, and if that's a problem, send it after 9.8 seconds, or some other interval. Read in small chuncks if that's required. Or just use a separate process.

But can two concurrent processes talk from the same local port concurrently?

Whether or not that's possible is irrelevant (however, this is UDP we're talking about, it's isn't about armies of ogres having to exit a tower through a narrow gate), as that is not what the OP seems to be doing. To quote him:

The client sends the first heartbeat from ports: 53036, 53037, 53038, 53039 to ports 8020, 8019, 8008, 8003. This repeats in every 10 seconds.

Oh, and do note that in my sketch, there's just one program anyway.