Yes. Absolutely reinforce that HQ first turn. It allows you to also reinforce other units to max levels. Once the engineer arrives, plan where you want the forts and let it build them, keeping in mind the delay and the expected time you think the enemy will appear. Having the engineer killed while building forts is not going to help the defense of Egypt. When happy with the front line forts, allow the engineer to start building a second line of forts east and west of the Suez Canal just in case. There is not much use for the Engineer elsewhere so you might as well make it work while you wait.
Once the front-line forts are built, man them and leave the troops alone to let the entrenchments build up. I've found that under these conditions and with a close HQ, it is very expensive and time consuming for the Axis to break the lines if they are not built to far forward. While the Axis batters on the forts, reinforce or rotate units and build up with others from UK (if no Sealion). The NZ and Coups will arrive via a DE and other units will auto deploy based on conditions. Reinforce and update all units so you are ready to move West when you sense weakness. Keep the RN ships busy by bombarding units along the shore. The effort may see wasteful, but if you cause some damage every few attempts, it adds up fast. Reinforce the ships as required to level 8 (maximum in East Med because of ports) and keep the carrier fully stocked with planes. Use it to either strike exposed units or better, to recon hexes to find units to strike with tactical bombers (and fighters) that were moved to Egypt. The air units are not needed in the UK for a while unless Sealion is done and if it is, your focus is probably not going to be Egypt anyway.