Some comments...
1. Romanizied "Mother" should read "okaasan" instead of "okasan"
2. For the "Dozo yoroshiku" part, the tone of speech (口調, くちょう) sounds a little strange, usually I would say "dozo yoroshiku onegaishimasu" (a more polite way of saying) as this situation might be too casual for the initial meet-up between Anne and the mother. Meanwhile, I think it is a little too casual for a conversation between an adult (Anne) and a kid (I am assuming Takeshi and Jun are both school kids).
3. "Takeshi", "Reiko" are common names, and there are quite a number of kanji's equivalent to the name, so for the sake of simplicity, only hiragana is used here.
Disclaimer: I am not native :p So, there might be mistakes.
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Reiko: Ano ( a pause word)...an Jenkins(u)-san desu ka
れいこ:あの、アンジェンキンスさんですか?
Girl: Iie...
女の子:いいえ。
Reiko (embarrassed) A! Sumimasen.
れいこ:あっ、すみません!
Takeshi: (also emnarrassed) Okasan!
たけし:お母さん!
Anne: Sumimasen, Yamaghuchi-san desu ka
アン:すみません、山口さんですか?
Reiko: Hai, so desu!
れいこ:はい、そうです!
Anne: Hajimemmashite, An Jenkinsu desu. Dozo yoroshiku (she bows)
アン:はじめまして、アンジェンキンスです。どうぞよろしく。
Reiko: (bowing) Hajimemashite, Yamaguchi Reiko desi. Dozo yoroshiku.
れいこ:はじめまして、山口れいこです。どうぞよろしく。
Reiko: Kore wa Takeshi desu. Musuko desu.
れいこ:これはたけしです。息子です。
Takeshi (bowing) Dozo yoroshoku.
たけし:どうぞよろしく。
Anne: (bowing) Dozo yoroshiku.
アン:どうぞよろしく。
(Takeshi bumbs into a schoolfriend Jun)
Takeshi: Konnichiwa!
たけし:こんにちは!
Friend: Konnichiwa!
友達:こんにちは!
Takeshi: Kochira wa An Jenkinsu-san desu.
たけし:こちらはアンジェンキンスさんです。
Friend: Hajimemashite. Suzuki Jun desu. Dozo yoroshiku ( bows)
友達:はじめまして、鈴木じゅんです。どうぞよろしく。
Takeshi: Ja mata ne!
たけし:じゃまたね!
Jun: Mata ashita!
じゅん:また明日!