Delayed separation of the umbilical cord is pathognomonic for leukocyte adhesion deficiency (autosomal recessive). It's due to absence of CD18 antigen that's necessary for formation of integrins, the substances that are responsible for the second step of leukocyte migration: adhesion. Because the leukocytes cannot efficiently reach the site of insult, they cannot engulf the offending substance, and no pus will be evident. Also, as a compensatory mechanism, there may be excessive leukocytosis.
Chronic granulomatous disease is characterizied by the susceptibility of infection with catalase-positive organisms like S. aureus, Pseudomonas cepacia, Aspergillus, Nocardia, and Serratia marcescens.
leukocyte migration is defective and we know that in addition to inflamatory role leukocytes also have role in inducing healing process(secreting FGF and interleukins) so healing of cord delays resulting in delayed separation(normal time is 3-14 days with average of 10 days)
obviously in this disorder no pus formation because of same defective migration...
but keep in mind that lymphocyte arm is less affected because they also have CCR receptors who somehow compensate migration defect