I just realized that I have no idea how other romanization systems (besides pinyin) deal with 儿化 (er hua).
What would be the Wade-Giles or Zhuyin-Fuhao equivalent to 电影儿 (diànyǐngr)?
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Mr Giles' phrasebook uses -r, the same as Pinyin. The phrasebook was published after he created the Wade-Giles system, so presumably it uses this system.
(EDIT: I previously said "I think this might have been before the Wade-Giles system was formalized", but then found that the date of publication was 1901, while Wikipedia says Wade-Giles "was given completed form with Herbert Giles' Chinese–English dictionary of 1892")
This is not a complete answer, but the Wade-Giles example chart on this Chinese-language Wikipedia entry includes "kêrh" as a transcription of the rhotacized "哥儿." I do not know the general rule.
The GR (Gwoyeu Romatzyh) transcription system, which was used officially in China from 1928-1958, has complex rules for 儿化 spellings. I can't find a complete chart online, but, again, Chinese Wikipedia has some details. GR uses "l" to represent rhotacization, but the precise rules for how the resulting work is spelled differ depending on the syllable final and tone.
兒化 is the process of adding a 兒 at the end of words in some Notĥern Chinese dialects like the Beijing's one. There is no specific transcription for that other than adding a "r" in pinyin and a "ㄦ" in Zhuyin Fuhao. Although, in Taiwan, it's almost never seen since people don't do 兒化…
Specifically for Zhuyin Fuhao they add "ㄦ" as an erhua marker after the Zhuyin tone mark of the erhua-ized syllable e.g. 电影儿 is transcribed "ㄉㄧㄢˋㄧㄥˇㄦ". Usually this ㄦ is added with no tone mark (which in Zhuyin otherwise marks first tone) but some dictionaries will instead mark ㄦ with a neutral tone marker i.e. "ㄦ˙" instead. Note that the full syllable 儿 never by itself is pronounced in either the first tone or neutral tone so in either case there is no possible ambiguity with the full syllable ㄦ.
Surprisingly hard to find Yale Romanization erhua rules or even erhua examples online. Might need to go to library and physically look at old books published (by Yale Press) in the US before the 1980's.