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Trying to see if I am at all on the right track. Is the thematic word in 58a from a children's novel? Hope that doesn't constitute an spoiler.

Yes.

Thank you!

I'm struggling with 19d. There only seems one Swedish chemist to fit 19a, although I can't parse that ... I can think of two words that fit into 19d, but can't see how they fit together, or parse.
And am also defeated by 43a. The second word seems obvious, and I guess there's another obvious two-letter word to precede it, but I don't see how that corresponds to the clue. Any hints welcome.

re 19d

swedish chemist with equation is straight clue

abbreviation for arrived then theme word 3 letters then i the us

hope this helps

19a is a themed clue. Letters 4 to 6 are from the theme.
19d is also themed.

43a apparently is an ozzie saying, had to look it up. Yes it is the nursery rhyme option.

43 across: the name of someone who lost her sheep is defined in Chambers as "a quick look" (Australian). The two-letter first part is US slang for "man" and the remainder is defined by various online dictionaries as slang for "friends" (derived from "people").

Many thanks for all the help - that's brilliant.

I had the wrong Swedish chemist. Who knew?