Is “little” an appropriate translation for 小 in peoples ‘nicknames’?
No. This translation sounds quite strange, in fact - I don't get the impression that 小燕 is something that even Chinese newspapers** would write, and this kind of name is really reserved for acquaintances/friends.

There very likely isn't an exact English translation; a proper description of 小-- is something like a prefix attached to a surname to indicate affection/endearment for someone younger (similarly 老 is used for someone older). To show similar affection/endearment in English you would really just use their preferred name (whether that'd be their nickname, full name or something else), and the closest example to 小-- in the English-speaking world would be something like the nicknames given to people in Aussie English. From Wikipedia:


  *Litotes, such as "not bad", "not much" and "you're not wrong", are also used, as are diminutives***, which are commonly used and are often used to indicate familiarity. Some common examples are arvo (afternoon), barbie (barbecue), smoko (cigarette break), Aussie (Australian) and pressie (present/gift). This may also be done with people's names to create nicknames (other English speaking countries create similar diminutives). For example, "Gazza" from Gary, or "Smitty" from John Smith. The use of the suffix -o originates in Irish Gaelic[citation needed] (Irish ó), which is both a postclitic and a suffix with much the same meaning as in Australian English.*


To mention someone differently based on age is far more neutral and sometimes even a subtle insult in English, implying either the person is immature (younger) or old-fashioned (older). For example in Scots, you may call someone laddie or lass, or elsewhere among older folks you may mention someone in contrast to their father/children like the younger/elder (Mr.) Potter.

I would just translate it as Mrs Yan; or even better, give the first name instead, as English speakers generally are not acquainted with the fact that Chinese women do not take their husbands' last names when they marry, and the difference in last names may be a bit jarring to read. 



**The translation for 小燕 is there because this is a story piece of the struggles of an immigrant couple, and the word choice is important for the reader to emotionally connect to the couple. Using 小燕 makes sense in a Chinese context, but again in English you'd only use the preferred name.

***Diminutives are the key word to search for if you wish to find affection/endearment terms of address in other languages.