What causes feeling of feet falling asleep to be delayed?
This paper, while mainly dealing with chronic nerve compression, appears to explain the effects of nerve compression on the blocking the transmission of neural impulses (causing the "sleeping limb" syndrome).

This paper describes how the application of pressure on a nerve causes a miniature case of compartment syndrome, where the pressure causes the intraneuron pressure to increase, leading to paresthesia. Edema also occurs within the neuron, causing pain signals to be sent out. In the experimental rats which were unable to relieve this pressure, permanent nerve damage resulted. 

When the effects from the nerve compression exceeds a certain level, the pain/tingling will result in you releasing the pressure from the limb in order to prevent nerve damage. 

However, the nerves are still blocked from transmitting their impulses due to the  compression blockage described in Paper 1. When the blockage is resolved, the neurons are now capable of transmitting their pain signals, which continue for a period of time until the pressure within the limb falls sufficiently below the pain threshold.