I was being conservative. The plan should have been, "How to take Khartoum, Port Sudan and the important railway junction at Atabara on Day 0." I'm not saying that all 3 objectives would have been taken, but they should have tried.There were several Sudanese battalions who, along with reinforcements from Egypt, would have eventually slaughtered the paras if left alone. Plus, Karthoum would require building up vulnerable airstrips in S-E Libya.
HOWEVER, it is my opinion that a determined-enough attack out of East Africa into Sudan might have led to a Sudanese mutiny, especially if Port Sudan is taken early (it got raided by cavalry the Italians somehow snuck through the desert OTL)
The garrison of the Sudan was 9,000 men (British and Sudanese) IIRC. This included 3 British infantry battalions, one of which was Khartoum, one was at Port Sudan and the third was covering the important railway junction at Atabara. IIRC the Sudanese troops were defending the frontier.
IIRC reinforcements didn't arrive until September when an Indian infantry brigade landed at Port Sudan. This was eventually built up into a force of 2 divisions. I doubt that it will be possible to reinforce the Sudan faster than that IOTL.
AIUI the vulnerable airstrips in S-E Libya existed IOTL. The Italians used them to fly supplies to Ethiopia IOTL. This included dismantled Fiat C.R.42 fighters in their S.M.82s. So I think that it would be feasible to take Khartoum and Atabara by an air landing from Libya and to fly reinforcements in afterwards. They don't have enough paratroops to take both at the same time IOTL, but I do think that they could take one or the other if they can organise it quickly.
The distances from Ethiopia to Khartoum, Port Sudan and Attabara were great, but a land attack was feasible due to the disparity between the forces of OTL. That is provided it can be launched before the rainy season that usually lasted from late June/early July until the end of September.
This explains the situation in June 1940 and the feasibility of an Italian invasion of the Sudan in June 1940 better than I can.
http://www.ibiblio.org/hyperwar/UN/UK/UK-Med-I/UK-Med-I-9.html