@Necropolitan not trying to be snarky here, but you may need to go back and watch it a third time.
While I will agree with you that the set up was not as clean as it should be (last week was definitely a brain-twister), they did eventually provide exposition through flashbacks.
Right at the end of the episode when the Doctor asks Belinda to remind him about Poppy (as they are talking in the living room after he is hit on by Belinda's mother), the reminding is made up of flashbacks to many of the other episodes this season. Each of those flashbacks starts with Belinda reminding the Doctor that she needs to be back by May 24 ,,, they then go on to provide more context than we originally saw.
That context is she needs to be back to her parents house to take over watching her own child since she is on nights at the hospital and her parents usually watch, but in this case her parents are off to a music event so she has to be back in time (no pun), no exceptions.
This also matches up with other scenes during this season where Belinda was shown back in the hospital where she worked and she was visibly frustrated when not getting off work on-time, or where she was shown rushing out of work. We had no context about that at the time (aside from perhaps being a nurse holding things together when the doctors overlooked things, aka the stereotype of nurses being the backbone of any heathcare system). Now we are provided context.
Even the Doctor being Poppy's father during the wish phase makes more sense after we have full context (and I did not think about this immediately, but it came together while formulating my thoughts for this message). Conrad based his wish world on his version of an ideal world, which was some sort of 1950s paradise (sort of) with defined gender roles, etc. Belinda as an unmarried single mother cannot fit into that model, she must be paired up. Conrad has no clue about her original partner, but does know she travels with the Doctor, so of course the Doctor is now her husband and Poppy's father. Both the Doctor and the Rani jump to the wrong conclusion about Poppy's origins because they cannot remember (the Doctor) or don't know/don't care (the Rani) about Poppy's original history.
Is it too convoluted? Perhaps.
Too clever by half? Probably.
But if you step back and reflect it does fit together and make a coherent whole with some rough edges.
Now with that all said, I would say it could and should have been cleaned up a bit.
Cheers
