This question has been asked and answered in a slightly different form at How able were pilots of three-person cockpits at running the normal FE work?, and I refer you to that for basic info. Concerning the detail in your question:
What difficulties would the pilots face for the remainder of the flight, including navigation and landing, with no one attending the flight engineer station in the cockpit?
It would be a nuisance for sure, but as long as there were no abnormal or emergency situations (other than the FE's personal situation), a nuisance would be all that it was. However, one could dream up scenarios that could get really sticky without the specialized knowledge that the FE would be expected to have that the pilots might not have (or had forgotten).
For example, would one of the pilots get up and sit at the FE station, leaving the other pilot the only person at the controls? Or would the person get up, go to the FE station to read or adjust something, then sit back at the pilot's seat? Or do they not have to move at all?
The Captain can see the FE panel. Some of the gauges along the aft portion would be a little hard to read. For example, if he wanted to see what the exact pressure of the Potable water was rather than just that there was pressure, he might have to get up to get close enough. Or if a generator fault light came on, he'd probably have to get up to see what the fault was.
There wouldn't be any need to stay in the FE seat. If the flight were long, there would be some trips to the FE seat for fuel management. You can have as many as 9 fuel tanks (7 is the usual), and there are rules for the fuel burn schedule. The most important and most time consuming FE task during cruise was keeping track of what the fuel burn was, especially on the 747-100 because it didn't have the fuel capacity of the -200. For example, a -100 going from JFK to Tel Aviv would sometimes not have the fuel to make it. Abeam Athens the FE would inform the Captain either than they had the fuel to continue or that they would need to divert to Athens for fuel.
If the Captain wanted to talk to company for some reason, it was usually the FE that established contact via HF, and it was usually the FE that answered SELCALs. But the pilots sometimes did it.
Flight attendants might show up and ask for a general cabin temperature change.
Of course, it the FE had a severe medical problem, you would probably land at the nearest suitable airport just as you would for a passenger medical emergency.
