Hello and welcome to rootschat.
You wrote
They sailed from Liverpool in Convoy WS1. A fast convoy of three ships comprising Queen Mary, Aquitania and Mauretania. I'm unsure of the actual date of arrival in Egypt.
Although records of troopships are always sketchy due to security concerns, It would seem AQUITANIA and MAURETANIA sailed from Liverpool 28/29/6/1940 and QUEEN MARY from Greenock on the same date.
After arriving in Freetown the Admiralty decided that rather than risk three large troopships in the Red Sea where the Italians had easy access to air bases, the vessels were diverted to Ceylon
arriving 29/July/1940
The troops were then transferred to North Africa via smaller vessels, some even traveling by train via India!
Read chapter 4 of the book "The Winston Specials" by Archie Munro available from Maritime Books Liskeard PL14 4EL or your local library. ISBN 13: 9781904459200 There are are several documents concerning Convoy WS1 in the Admiralty series in the British National Archive.
Your dads service record may tell you the date of his arrival in Egypt. When you finally receive it. Make a post on the WW2 board. There are several people there who will be able to interpret it for you.
I guess you have seen this.
https://discovery.nationalarchives.gov.uk/details/r/C16668267