Do British railways (post privatization) still use the "hook and chain" or screw couplings, and if so why?

There are many different types of coupler on the UK mainline network depending on the type of train.

The 3-link coupler went out ages ago (not sure when), but the screw coupler remains in use and is probably still the standard coupler. The advantage of the screw over the 3-link is that you can take out a lot of the slack between the vehicles which reduces the severity of a "snatch" (the jolt experienced as a train starts away and takes up the slack in the couplers) and the likelihood of a broken coupler. There are quite a lot of freight vehicles that use the screw coupler. Although some of the newer types of freight wagons now use a buckeye (knuckle...?) coupler, the standard way of coupling the loco will be using the screw coupler as that is the standard drawgear fitted to UK loco classes.

Loco-hauled passenger stock has, for some considerable time, used a version of the buckeye coupler known as a drophead. Where a coach is coupled to another coach the buckeye coupler will be in the raised position and in use. However, at the outer ends where the loco will be attached, the buckeye is dropped to reveal a coupling hook to which the loco's screw coupler can be attached in order to draw the train. Modern passenger locos, such as the Class 90 and 91 electric locos, have themselves been fitted with drophead buckeyes so that they can couple to passenger trains this way. However, at the outer ends of the train there is still the need to fit standard drawgear so that the train can be rescued by a conventional loco in the event that it fails.

Where things get complicated is with modern multiple-unit (MU) trains. Back in the 1950's and 60's these were generally constructed with either standard drawgear or a buckeye coupler. However, these types have all since been withdrawn and replaced with modern types that use one of three different types of autocoupler that provides both the physical connection between MU's together with the air and electrical connections that are needed for MU's to operate in multiple with each other. While these can be a bit temperamental from time to time, they are much quicker and easier to use and remove the need to get down between the MU's to hook up the air hoses and multiple-working jumper cables. Where they are not so good is if an MU needs rescuing by a loco (e.g. an electric MU "off the juice"), as you will need an adapter in order that the loco can be attached.

The other major type of coupler used in the UK is the bar coupler. These are generally found between vehicles formed into a permanent or semi-permanent set. This could be between the vehicles of an MU or certain types of freight wagon. These are not uncoupled in service but may require to be split for maintenance purposes.

I hope this helps. If you need more detail or clarification, just let me know.

"Buffer & Chain" is the standard method of coupling all over Europe. There have been efforts to introduce an automatic coupler (an advanced version of the russion SA-3 coupler) but this is very hard because there are hundreds of railway companies in Europe and the adaption of all rolling stock would take a long time and cost a lot of money.

There are however automatic couplers in use in some areas:

Dedicated trains of large coal-mining companies (e.g. in germany) have their own rolling stock equipped with SA-3 couplers - simply because chain couplers are not capable to withstand the heavy weight of the trains).

Multiple Unit passenger trains mosly use Scharfenberg couplers (which are also able to be enhanced with automatic coupling of power and control cables). Scharfenberg couplers however are only suitable for powered vehicles since they need to be heated to avoid icing up in winter (or they need to be covered, which involves human intervention to remove the cover when couping, resulting in an only 'semi-automatic' coupler).