No doubt her first cousins such as the Kents,Gloucesters and the Hon Mrs Rhodes still in private give her the affectionate nickname.Would think too that Margarethe of Denmark and Constantine and Anne-Marie of Greece would be among the selected few who are allowed this intimacy.
In addition:
The King of Norway and his sisters have always used Lilibet when talking to her. It seems likely Queen Sonja also calls The Queen Lilibet. I heard that Martha-Louise uses Lilibet, so I would assume Crown Prince Haakon does as well. No idea about the Crown Princess of Norway.
Crown Prince Alexander calls her Lilibet. King Michael, Queen Anne, and their daughters also use Lilibet.
Countess Mountbatten and Lady Pamela Hicks address The Queen as Lilibet in private. As does The Duke of Fife. None of their spouses used such familiarity, although Lord Braburne called the Duke of Edinburgh Philip.
Princess Michael of Kent calls The Queen only Ma'am, even in private and it's possible the same goes for The Duchess of Kent and The Duchess of Gloucester.
I recall reading somewhere that the late Princess Juliana of the Netherlands called The Queen Lilibet, but that Beatrix addresses her as Elizabeth.
There was a private party a couple of years ago and among the guests were The Queen, the Duke of Edinburgh, and a Romanov prince. I can't recall which one it was. (Nicholas Romanovitch??) During an interview since that time, the Prince was asked how he addressed The Queen and her husband to which the Prince replied, "As Ma'am and Sir, of course." The Prince said that he is on very social terms with Prince Michael of Kent whom he addresses as "Michael".
I do not remember the reason or the nature of the interview, but perhaps someone here might. Marlene seems to be more up on such things.