Digital TV tuning is slow because of compression; when you tune to a stream in progress, you usually can't meaningfully decode it until you receive a I-frame. But I-frames are bigger than motion predicted frames, so it makes sense to only ocassionally send I-frames. Latency: technically unbounded, usually a couple of seconds.
It potentially gets a bit worse if you're on a switched video cable system where your box has to request channels, as now there's a request / response delay. But it shouldn't be too long for local comms... And in theory, the head end could start the stream with an I-frame (otoh, it may not have the processing power to decode/encode).
In theory, a TV with multiple tuners could do predictive decoding to help with channel surfing, but I don't think anybody actually does that.