>I know very little about UK property issues
Don't let that stop you!
>but would speculate that any housing shortages are caused by regulations.
Which regulations? The UK rental market is almost entirely unregulated. It is less regulated than the rental market in many US states/cities, for example. Landlords can charge whatever rent they like, evict people for no reason, and put up rents without limit when renewing rental contracts.
If any regulations are to blame here, they're likely to be the regulations that prevent building new houses in green belt areas, not regulations on rentals.
>Any increase in expenses will need to be generated through the rent of the property.
You say this as if it's obviously inevitable, but of course one possibility is that landlords will simply make a bit less money on rentals than previously. This is why regulations do have something to do with it: if there are limits on how much landlords can raise rents, then they can't simply defray any increased expenses via arbitrary rent increases. At present, however, there are no such limits.