EgyptAir /Etihad interline/link bookings?
 

I have purchased two separate Egyptair tickets:
1. MEL-CAI, arriving at 06.40 am;
2. CAI-ATH departing at 09.30am on the same day.

The first leg of the MEL-CAI flight, MEL-AUH is operated by Etihad
I have asked Egyptair (both by phone to their Australian office and by email to their head office) if they can link the bookings so I can check my luggage all the way through to ATH, but they have said they cannot, and it will depend on whether the Etihad check in clerk in MEL is prepared to.

Any reason Egyptair refuses to link the booking?

Given the airlines codeshare, and Etihad is already checking the bags on to Egypt Air for the AUH-CAI leg, I am hoping they will have no problem doing so for the separate Egypt air flight to ATH.

Has anyone had any experience doing similar, or have advice?

if I have no luck, will I have enough time to purchase a visa on arrival, clear immigration and customs, collect my bags and check in again?

You cannot link/combine/join/merge/stick together two separate itineraries (PNRs).
That's why MS is not doing anything.
Some airlines will put a message in the file noting two separate itineraries ... but this has no practical impact.
As MS has already told you, the EY check-in agent in MEL will tell you whether he/she can/will check through your bag(s).
Many carriers no longer check-in luggage for passengers on separate itineraries - like you - particularly involving carriers not in the same alliance. So YMMV.

From whom did you purchase the first ticket? Ask them to add the CAI-ATH segment as a ghost/passive segment into the PNR. That should enable the MEL checkin agent to check your luggage thru.

BEWARE MS is denying checking bags through to their final destination on two separate tickets:

I have extensive experience for the past 5 years in routing travel from the Gulf through CAI on two separate tickets connecting from MS to TK and LH. Last month, for the first time in 5 years MS refused to check my bags through on a separate ticket to LH connecting in CAI. I had a 4 hour layover in CAI so there was no short connection issue. The Ticket agent at the originating Gulf Airport (the same where I had done this with 100% success for 5 years) said that as of August 2018 they could not check bags through on separate tickets, even those purchased through Egyptair and connecting to other Star Alliance Carriers, like LH However, once in CAI, I took my claim checks to the Transfer Desk and asked if they could some how check my bags through without me buying a $25 visa, claiming my bags and re-checking them with LH. The MS Transfer Desk agent took my bag claims, retrieved and brought me my bags and the LH Flight representative who checked my bags on to their USA destination. I had to wait 2 hours in Limbo, but it did finally work.