In the past decade, superconducting nanowire single-photon detectors (SNSPDs) have gradually become an indispensable part of any demanding quantum optics experiment. Until now, most SNSPDs have been coupled to single-mode fibers. SNSPDs coupled to multimode fibers have shown promising efficiencies but have yet to achieve high time resolution. For a number of applications ranging from quantum nano-photonics to bio-optics, high efficiency and high time resolution are desired at the same time. In this paper, we demonstrate the role of polarization on the efficiency of multimode-fiber-coupled detectors and fabricated high-performance 20 µm, 25 µm, and 50 µm diameter detectors targeted for visible, near-infrared, and telecom wavelengths. A custom-built setup was used to simulate realistic experiments with randomized modes in the fiber. We achieved over 80% system efficiency and <20 ps timing jitter for 20 µm SNSPDs. Also, we realized 70% system efficiency and <20 ps timing jitter for 50 µm SNSPDs. The high-efficiency multimode-fiber-coupled SNSPDs with unparalleled time resolution will benefit various quantum optics experiments and applications in the future.