Are there nerves in the umbilical cord?
Short answer
There are no functional nociceptors present in the umbilical cord.

Background
Having had the opportunity to cut the umbilical cord of one of my kids, I can anecdotally confirm neither newborn nor mother gave any sign of pain during the procedure. That is, when cutting the umbilical after the birth of a full-term baby. The only person clenching his teeth was dad :)

In fetuses at gestational age of 18-36 weeks no stress response was noted after puncturing the umbilical (Teixeira et al., 1996), indicative of the absence of pain receptors connected to the baby's brain. 

I couldn't find information about pain in the mother, but I cannot imagine that the maternal central pain centers are being connected to the umbilical, which would be pretty extraordinary. This, because long-distance neural connections are rarely, if ever, established in adult humans. Also, I haven't heard any reference to the mother feeling pain because of injury to the umbilical cord. For example, knots in the umbilical cord, or cords wrapped around the fetus' neck due to excessive movements of the fetus are never noticed by the mother, and are only revealed during ultrasounds.   

Note, however, that the umbilical cord is innervated by cholinergic neurons that are thought to be involved in the control of blood transfusion to the fetus, but presumably do not carry sensory information to either fetus or mother (Ellison, 1971). 

References
- Ellison, Am J Anat (1971); 132: 53-60
- Teixeira et al., The Lancet (1996), 347: 624