Do they still play croquet at the 'All England Lawn Tennis and Croquet Club', aka Wimbledon?

They do. Interestingly, they did not during large periods of the club's existence.

When the club was founded in July 1868 as "The All England Croquet Club" they didn't even have ground to play - the championships were held in Crystal Palace.

In 1877 the name was changed into "The All England Croquet and Lawn Tennis Club" but the interest in Croquet declined with the years. By 1882, croquet was not played anymore and the word was removed from the club's title.

In 1899 the Club Championship was hed again and the word "croquet" was restored to the club's title. However, the interest didn't last long and after the 1904 Championship the game faded again. This time the club retained the same title.

For the next fifty years croquet was not provided in the club. It returned on 7 September 1953 when a match was arranged with the Croquet Association to mark Coronation Year. The current full size lawn is located in the Southlands College ground.

https://www.wimbledon.com/en_GB/atoz/about_aeltc.html

They played croquet at Wimbledon (on a half-size court) until 2007, when new Court No. 2 was built on top of it. The croquet court (full size) was created in 2007 at Southlands College, 3 miles by road northwest of Wimbledon. Each summer the croquet court becomes two tennis court (practice courts for Wimbledon). Two tennis courts fit neatly into a croquet court.
https://www.watfordcroquet.org.uk/curiosities.php
https://www.bbc.com/news/uk-england-gloucestershire-13930153

From 1957 to 2007, the croquet lawn (not full-size) was situated at the southern end of the Grounds, alongside the Church Road boundary fence. The Club Croquet Championship was reintroduced together with a Handicap singles event in 1960 and in recent years a golf croquet event was started. In the autumn of 2007 a new full-size lawn was created in the Southlands College ground, which was opened on 19 April 2008 by Professor Bernard Neal, President of the Croquet Association. Professor Neal was the Club champion 38 times between 1963 and 2005.

https://www.wimbledon.com/en_GB/atoz/about_aeltc.html

Croquet court at Southlands College (google maps) - https://tinyurl.com/croquet-court-southlands
Distance Wimbledon to Southlands (also google maps) - https://tinyurl.com/wimbledon-to-southlands