There aren't many feature films, let alone comedies, about learning English, but the Bollywood film English Vinglish is a recent one that has received good reviews (and which I will go and see if and when it comes to a local cinema - I think I might have missed it when it came out about a month ago). It's about an Indian woman, teased by her family about her poor English, who enrols on a language course while visiting New York to stay with relatives (watch the trailer here).
I'm not exactly sure about the significance of the title and, in particular, Vinglish. It could be that Hindi forms reduplicative words (examples of reduplicative words in English include nitty-gritty, hocus-pocus) using the letter V for the second element (and this Wikipedia page suggests it does). Or, the vin of Vinglish could be a play on the word 'win'.
English as spoken by speakers of Hindi is sometimes called Hinglish. There are other words for other varieties of English spoken elsewhere, which are imbued with vocabulary and features of the home language, including Thaiglish (Thailand) and Konglish (South Korea). Those last two examples are not in the OED, but other varieties of English, or varieties of other languages heavily influenced by English, are. They are listed below with their OED definitions:
Chinglish: A mixture of Chinese and English; esp. a variety of English used by speakers of Chinese or in a bilingual Chinese and English context, typically incorporating some Chinese vocabulary or constructions, or English terms specific to a Chinese context.
Hinglish: Esp. in India: a mixture of Hindi and English; esp. a variety of English used by speakers of Hindi (or in a bilingual Hindi and English context), characterized by frequent use of Hindi vocabulary or constructions.
Japlish: A blend of Japanese and English spoken in Japan: either the Japanese language freely interlarded with English expressions or the English language spoken in an unidiomatic way by a Japanese speaker. (This is the oldest entry among this list - it went into the OED in 1976 and hasn't been updated since. Spanglish and Yinglish went in in the 1980s, and the others only within the last ten years.)
Singlish: An informal variety of English spoken in Singapore, incorporating elements of Chinese and Malay. (There is also another entry for Singlish: An informal variety of English spoken in Sri Lanka, incorporating elements of Sinhala.)
Spanglish: A type of Spanish contaminated by English words and forms of expression, spoken in Latin America.
Yinglish: A jocular name for a blend of English and Yiddish spoken in the United States; a form of English containing many Yiddishisms