Travel to India and Singapore after British passport

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I recently received my British passport and have no time to surrender my Indian passport before I travel this week. I have read several threads on this site that indicate it's okay to travel with both passports during the 90 day grace period for keeping Indian passports (some of the feedback has been incredibly useful!). Most threads seem to agree that I can do this:

1. Leave UK: Indian passport
2. Enter India: Indian passport
3. Leave India: Indian passport
4. Enter UK: British passport through e-gates

However my plan has some minor variations - this is what I plan to do:

1. Leave UK: Indian passport
2. Enter India: Indian passport
3. Leave India: Indian passport
4. Enter Singapore: British passport (first use of new passport)
5. Leave Singapore: British passport
6. Enter UK: British passport through manned gates (as travelling with child)

Logic says this shouldn't be a problem but I was wondering if Singapore might raise any issues coming in with a fresh new passport (would they know there is no exit data from the UK using it?). Also, hope the manned gates are still fine to enter back into the UK without issues.