  We study the leptoquark model of Buchm\"uller, R\"uckl and Wyler, focusing on
a particular type of scalar ($R_2$) and vector ($U_1$) leptoquark. The primary
aim is to perform the calculations for leptoquark production and decay at
next-to-leading order (NLO) to establish the importance of the NLO
contributions and, in particular, to determine how effective the
narrow-width-approximation (NWA) is at NLO. For both the scalar and vector
leptoquarks it is found that the NLO contributions are large, with the larger
corrections occurring for the case vector leptoquarks. For the scalar
leptoquark it is found that the NWA provides a good approximation for
determining the resonant peak, however the NWA is not as effective for the
vector leptoquark. For both the scalar and vector leptoquarks there are large
contributions away from the resonant peak, which are missing from the NWA
results, and these make a significant difference to the total cross-section.
