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Always-on TinyML perception tasks in IoT applications require very high energy efficiency. Analog compute-in-memory (CiM) using non-volatile memory (NVM) promises high efficiency and also provides self-contained on-chip model storage. However, analog CiM introduces new practical considerations, including conductance drift, read/write noise, fixed analog-to-digital (ADC) converter gain, etc. These additional constraints must be addressed to achieve models that can be deployed on analog CiM with acceptable accuracy loss. This work describes $\textit{AnalogNets}$: TinyML models for the popular always-on applications of keyword spotting (KWS) and visual wake words (VWW). The model architectures are specifically designed for analog CiM, and we detail a comprehensive training methodology, to retain accuracy in the face of analog non-idealities, and low-precision data converters at inference time. We also describe AON-CiM, a programmable, minimal-area phase-change memory (PCM) analog CiM accelerator, with a novel layer-serial approach to remove the cost of complex interconnects associated with a fully-pipelined design. We evaluate the AnalogNets on a calibrated simulator, as well as real hardware, and find that accuracy degradation is limited to 0.8$\%$/1.2$\%$ after 24 hours of PCM drift (8-bit) for KWS/VWW. AnalogNets running on the 14nm AON-CiM accelerator demonstrate 8.58/4.37 TOPS/W for KWS/VWW workloads using 8-bit activations, respectively, and increasing to 57.39/25.69 TOPS/W with $4$-bit activations.