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I'm in a similar situation. I plan to apply via the NCS for naturalisation and a UK passport simultaneously. I need to make some trips abroad, and I need to time this application so I won't be out of the country when they issue the passport and risk trouble.

Am I right in thinking that what I'll receive first will be the invitation to book the citizenship ceremony? So if this came when I was out of the country, I could just reenter the UK on my original birth-country passport (USA), right?

And then after that, I have to book the ceremony, then give back my BRP after it and wait for my UK passport, right? During this time, if I travelled to the Netherlands (necessary for work, can't be rearranged), could I reenter on my US passport? Or would I get in trouble at the border? If so, what would happen? Refusal of entry?

Many thanks! I'm on endless hold on the phone right now, but so far no agency has answered this for me.

Am I right in thinking that what I'll receive first will be the invitation to book the citizenship ceremony? So if this came when I was out of the country, I could just reenter the UK on my original birth-country passport (USA), right?

You are ONLY British once you have attended the ceremony. The approval letter/ceremony letter is just that, a letter. Most members have to call and book the ceremony so it would be wise to have someone keep an eye on your mail in case you are out of the UK. You have 21 days to book the ceremony and 90 days to attend the ceremony.

And then after that, I have to book the ceremony, then give back my BRP after it and wait for my UK passport, right? During this time, if I travelled to the Netherlands (necessary for work, can't be rearranged), could I reenter on my US passport? Or would I get in trouble at the border? If so, what would happen? Refusal of entry?

You could try but it might take a while for them to check your citizenship status. Assuming you have applied for the joint citizenship and passport scheme, you can carry your naturalisation certificate to ease your entry back into the UK.