High-temperature bivariate expansions have been derived for the two-spin-correlation function in a variety of classical lattice $XY$ (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimensions $d=1$, 2, and 3, are expressed in terms of the two variables ${K}_{1}={J}_{1}/kT$ and ${K}_{2}={J}_{2}/kT$, where ${J}_{1}$ and ${J}_{2}$ are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This paper deals in particular with the properties of the $d=3$ uniaxial $XY$ model (ANNNXY model) for which the bivariate expansions have been computed through the 18th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter $R={J}_{2}/{J}_{1}$. Universality with respect to $R$ on the critical line separating the paramagnetic and the ferromagnetic phases can be verified, and at the same time the very accurate determination $\ensuremath{\gamma}=1.3177(5)$ and $\ensuremath{\nu}=0.6726(8)$ of the critical exponents of the susceptibility and of the correlation length, in the three-dimensional $XY$ universality class, can be achieved. For the exponents at the multicritical $(m,d,N)=(1,3,2)$ Lifshitz point the estimates ${\ensuremath{\gamma}}_{l}=1.535(25)$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=0.805(15)$, and ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=0.40(3)$ are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.