Is it customary to call the former President George W. Bush “W.”, and Mrs. Bush “Bar”, in public?

Solution 1:

“W.” is quite common, but probably mostly because his father – who also served as U.S. President – had the same first and last name, so they often get differentiated by their middle initials.

JFK, FDR, and LBJ are all commonly-used monikers for those presidents. It's worth noting that the U.S. had already had a President Roosevelt and President Johnson when Franklin and Lyndon took office, so maybe that was a contributing cause.

As for “Bar”, we often use “Barb” (or sometimes “Babs”) as a shortened form of Barbara, but I've never seen the former First Lady referred to as “Bar” until you referenced it in this question.

The “Silver Fox” nickname is a reference to her platinum gray hair, which was prominent even when her husband was in office; she never bothered to color it, and being in the public eye wasn't going to change that. From Wikipedia:

...her interest in domestic staples such as church, gardening, and time spent with family while placing less emphasis on style sense and fashion; she drew attention to both her famous white hair and disinterest in wearing designer clothes.

According to some sources, her hair lost its color due to stress and grief over the loss of a child:

In 1953, the Bushes' daughter Robin died of leukemia. It severely affected Barbara Bush, and is the incident that is credited with beginning to turn her hair from a light brown color to chalk-white.

Solution 2:

A scholarly study addressing some of your questions is available online: Anna Gladovka, ‘The Semantics of Nicknames of the American Presidents’, from the Proceedings of the 2002 Conference of the Australian Linguistic Society.

Prof. Gladovka surveys presidential nicknames all the way back to Washington (‘the Father of His Country’). Among these are ‘James the Lesser’ (Monroe), ‘Ten-cent Jimmy’ (Buchanan), ‘Handsome Frank’ (Pierce), ‘Honest Abe’ (Lincoln), ‘Big Bill’ (Taft), ‘Ike’ (Eisenhower),‘Tricky Dicky’ Nixon) and ‘Slick Willie’ (Clinton).

Of particular relevance is this passage:

3.3. Initials used in nicknames
Three of the presidents were commonly referred to by their initials: Franklin Delano Roosevelt (FDR), John Fitzgerald Kennedy (JFK), Lyndon Baines Johnson (LBJ). These presidents possessed high degree of popularity from the public for their political achievements. These short forms of reference became widely used, and it is difficult to find cases of the use of this form with a negative determiner. So, it rather would be used positively.

When a speaker uses initials or abbreviations s/he implies that the audience is aware of what these letters stand for and shares a similar attitude towards the “encoded” person.

 (5) The expressive value of nicknames with INITIALS (JFK)

   (a) I want to speak about X the way people speak about someone if they think about      this person like this:
   (b)   all people in this country know this person
   (c)   I am one of these people
   (d)   if I say “JFK”,
   (e)   all people in this country will know the person I am thinking of
   (f)   I don’t have to say more
   (g)   I know that many people feel something good towards this person

The most common nickname of the present US President George W. Bush – W or Dubya (as pronounced by him with Texan accent) does not fully suit this category. It uses the initial for his middle name to refer to his Texan origin and the fact of his being the son of the forty-first president George Bush. So, there is more meaning in it than in the described nicknames. It is not as positive as JFK, for example, but still quite friendly and indulgent.