We analyze the impact of nonlocality on the waveguide modes of metallodielectric multilayers and optical patch antennas, the latter formed from metal strips closely spaced above a metallic plane. We model both the nonlocal effects associated with the conduction electrons of the metal and the previously overlooked response of bound electrons. We show that the fundamental mode of a metal-dielectric-metal waveguide, sometimes called the gap plasmon, is very sensitive to nonlocality when the insulating, dielectric layers are thinner than 5 nm. We suggest that optical patch antennas, which can easily be fabricated with controlled dielectric spacer layers and can be interrogated using far-field scattering, can enable the measurement of nonlocality in metals with good accuracy.