The Heir himself would have been more used to hearing within the family circle the standard Russian diminutives for 'Alexei' (Aleksei, Alexey):
Alyosha, Alyoshka, Alyoshenka, Alyoshechka; Lyosha et al as well as just plain Lyosh; Lyolya; Alyochka; and there is also a derogatory form I cannot imagine being applied to the Tsarevich; Lyoshok.
The use of 'Alex' in Russia specifically as a short form of Alexei was not unknown in the 1940's.
He was more apt to be addressed as 'Alexei Nikolaevich' by those given permission to do so, presumably pronounced 'Alexei Nikolay'ich'.
'Lex' is modern, affectionate in its own way, and even sweet, but the Heir himself would never have imagined such a nickname.
Dubbing the youngest member of the family 'Baby' is a common Russian practice, however -- it is not unheard of for folks in their twenties to be referred to in those terms to this day, so Alexei Nikolaevich was not alone. Can anyone actually cite an historical reference to the Tsarevich not liking the nicknames his parents gave him?