What date(s) are you interested in?
From 1811 to 1860 I can find plenty of mention of "committed to Warwick House of Correction", often sentenced with "hard labour".
In 1860 the sale of County Prisons was advertised ie "The Justices of the County of Warwick are prepared to receive tenders for the purchase of the buildings comprising the old house of correction belonging to the County, situate in the Borough of Warwick, together with the site, containing an area of 3,300 square yards"
The last commitment to this institution I can find is 1863.