Half-life and production rate of ion channels
The rate varies with channels and channel subunits, and you can also distinguish between membrane half-life and degradation half-life, because channels can be cycled bidirectionally between the plasma membrane and internal membranes like the ER. For many (most likely all??) channels these processes are under the control of phosphorylation and/or ubiquination.

For a rough estimate of magnitude though, the relevant time window is hours to days. I'll include a couple references below to support this time frame.

References



Bedford, F. K., Kittler, J. T., Muller, E., Thomas, P., Uren, J. M., Merlo, D., ... &amp; Moss, S. J. (2001). GABAA receptor cell surface number and subunit stability are regulated by the ubiquitin-like protein Plic-1. Nature neuroscience, 4(9), 908-916.

Colley, B. S., Biju, K. C., Visegrady, A., Campbell, S., &amp; Fadool, D. A. (2007). Neurotrophin B receptor kinase increases Kv subfamily member 1.3 (Kv1. 3) ion channel half-life and surface expression. Neuroscience, 144(2), 531-546.

Huh, K. H., &amp; Wenthold, R. J. (1999). Turnover analysis of glutamate receptors identifies a rapidly degraded pool of the N-methyl-D-aspartate receptor subunit, NR1, in cultured cerebellar granule cells. Journal of Biological Chemistry, 274(1), 151-157.