その上 details

Examples

on top of that; and besides — a further circumstance piled onto the one just described

昨日きのう、財布さいふを落おとしてさ。その上うえ、電車でんしゃまで止とまってて最悪さいあくだったよ。

kinou, saifu o otoshite sa. sono ue, densha made tomattete saiaku datta yo.

I dropped my wallet yesterday. And on top of that the trains stopped too — it was the worst. casual

この部屋へやは日当ひあたりがいいです。その上うえ、収納しゅうのうも十分じゅうぶんにあるんです。

kono heya wa hiatari ga ii desu. sono ue, shuunou mo juubun ni aru n desu.

This room gets good sunlight. On top of that, there's plenty of storage. neutral

本日ほんじつは雨あめの中なかをお越こしくださいまして、ありがとうございます。その上うえ、お土産みやげまで頂戴ちょうだいしまして、恐縮きょうしゅくでございます。

honjitsu wa ame no naka o okoshi kudasaimashite, arigatou gozaimasu. sono ue, omiyage made choudai shimashite, kyoushuku de gozaimasu.

Thank you for coming out today in the rain. And on top of that you brought a gift — I'm quite overwhelmed. formal

それに
それに is the lighter of the two and will tack on a passing afterthought; その上 says the second circumstance lands ON TOP of an already loaded first one, so what it adds has to weigh as much as what it is piled onto — which is why complaints and praise cluster on その上 and a throwaway extra takes それに.
しかも
しかも stamps the second fact as noteworthy in its own right and will follow with a bare fragment (三時間さんじかん待まった。しかも、外そとで); その上 reports that the circumstances have accumulated and wants a clause with weight of its own after it, so ?その上うえ、外そとで does not stand — both praise and complain equally well, and the fragment is what separates them.
に加えて
に加くわえて names its addition as a noun inside the sentence and reads as business writing (残業ざんぎょうに加くわえて休日きゅうじつ出勤しゅっきんも); その上 stands at the head of its own sentence with a full clause after it, and is ordinary in speech.

上で
One で decides the meaning — その上で means "having done that first, and on that basis" (資料しりょうは読よんだ。その上うえで決きめる), while その上 without で adds a second circumstance and sequences nothing; a reader who supplies the missing で turns an accumulation into a procedure.