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M dwarfs remain active over longer timescales than their Sunlike counterparts, with potentially devastating implications for the atmospheres of their planets. However, the age at which fully-convective M dwarfs transition from active and rapidly rotating to quiescent and slowly rotating is poorly understood, as these stars remain rapidly rotating in the oldest clusters that are near enough for a large sample of low-mass M dwarfs to be studied. To constrain the spindown of these low-mass stars, we measure photometric rotation periods for field M dwarfs in wide binary systems, primarily using TESS and MEarth. Our analysis includes M-M pairs, which are coeval but of unknown age, as well as M dwarfs with white dwarf or Sunlike primaries, for which we can estimate ages using techniques like white dwarf cooling curves, gyrochronology, and lithium abundance. We find that the epoch of spindown is strongly dependent on mass. Fully-convective M dwarfs initially spin down slowly, with the population of 0.2--0.3M$_\odot$ rapid rotators evolving from $P_{\rm rot} < 2$ days at 600 Myr to $2 < P_{\rm rot} < 10$ days at 1--3 Gyr before rapidly spinning down to long rotation periods at older ages. However, we also identify some variability in the spindown of fully-convective M dwarfs, with a small number of stars having substantially spun down by 600 Myr. These observations are consistent with models of magnetic morphology-driven spindown, where angular momentum loss is initially inefficient until changes in the magnetic field allow spindown to progress rapidly.