Hi Amythyst
Modern tufteds use a polypropylene primary backing, but may have a hessian/jute secondary backing. However, to make this shrink would need a lot of abuse and then it may not even shrink. Having said that, there are still some older tufteds around that have a hessian/jute primary backing, and these do shrink quite easily.
As for Axminsters and Wiltons (traditional wiltons or face to face) if they have a synthetic yarn they are much more prone to shrinkage. Classic examples are Acrylic pile ax's and face to face wiltons popular in the 80's. The reason is that synthetic yarns retain little moisture, so gravity and/or higher pressure application through a sprayer or wand can lead to moisture transfer to the backing yarns.
When we come to wool yarns, they will retain much more water than synthetic fibres, typically about 30% by weight (synthetics are typically between 0.04% and 5%). This retention property of wool keeps more moisture from the backing, but only initially. Mother nature designed vegetable fibres to absorb water very effectively in order to feed the plant. For jute and cotton, this can be upto 100% by weight. So for want of a better description, the jute backing yarns will try to "suck" water from the wool pile. This can lead to shrinkage.
When it comes to the varying degrees between a genuine wilton and a genuine axminster, the wilton will typically have about 30% more face yarn of wool over an axminster, so this would be like having a 30% larger reservoir of water. The wilton also has about 30% more jute than an axminster, so is working more to absorb the moisture too.
Hence my statement that an axminster may shrink and a wilton will shrink. But a correctly executed and appropriate clean on a correctly fitted carpet should in neither case present a shrinkage problem. The carpet will tighten on the grippers, but not shrink.
With wool carpets, there is a rule of thumb that to return to the dryness level of pre-clean, regardless of system used, will take about 48 hours.
Safe and happy cleaning

Ken