Hi OP,
I have an April DD starting reception too. Also I have an older child, and am finding that starting school is a whole different experience for second/further children than for PFBs!
You said the school doesn't have a January intake - well it might discourage this, but providing it is a state school in England you can simply send them an e-mail now or in September stating that your child will be starting in January (or April, for that matter) and there is nothing they can do about it.
Part-time is a little more complicated, in that you have a right to go part-time I believe, but the exact format of this has to be agreed with the school, so they might veto your desired way of going part-time. Even if you can legally enforce your own part-time pattern onto the school it makes a lot of sense to discuss and find an amiable solution with the school, that actually works for your child and for the school.
On the other hand, with an April baby you can stay part-time for the entire year if you wish, as your child won't be CSA until September 2019. But you must take up your place by summer term 2018 even though your child will not yet be CSA, or you will lose your place.
You asked for experiences:
A child in DS' class started in February after half-term break, at age 5. All other 29 children had started together in September, all full time from the start.
This child was very popular amonst the kids as the 'new child' and had no social issues whatsoever. Whereas in September, there was an incredible amount of social wrangling going on, establishing social hierarchies in ways I found shocking for 4/5 year olds, and was very painful for my own child.
Learning wise, the child was at least average in every topic, ahead in some (my own child's 'academic' learning slowed right down in reception, as he was one of 30 and preoccupied with social stuff rather than learning at his own pace, following his own interests, with close adult guidance/support - he would have learned much more in terms of maths/reading if I had kept him home).
Regarding settling in to the school routines and missing all the 'induction' activities at the beginning of the year, this child only profited. Instead of having to show 30 children the ropes, the teacher got to focus on just the one - and he had 29 class mates he could ask/copy if he was unsure. Whereas at the beginning of the year, everything can be a bit manic as none of the children know what's what, they copy each others' wrong behaviour, the teacher has to streamline everyone into a working routine before they can get to know and consider individual children's needs. For my DD, the first few weeks are what I am most concerned about, for this reason.
A couple of years later, a friend sent her summer born DD part-time. Initially she attended mornings only, after four weeks they changed to mornings+lunch, by October half-term the girl was asking to go full time, and though she did get pretty tired, never looked back. She definitely wasn't left out socially due to this. My take is that due to having her needs be met by having more time off, she was more able to pay attention/join in at the times she was in school. Also I think that for many children the social aspects and the very scarcely supervised free play in reception are the hardest, most tiring parts.
My now 8yo DS often said he didn't like nursery school - it turned out, because of meal times and the teachers making him drink more than he wanted to. In reception, he began really to dislike school, again because of lunch time: On the one hand, because there is preciously little supervision at lunch (nor in the times when most of the class has 'free play' whilst the teacher is working with a small group). If there happens to be a child in the class who for whatever reason lashes out at other children, verbally or physically, your child at age four will have to learn to deal with it pretty much on their own. You can 'tell the teacher' (if you recognise the behaviour as wrong in the first place - my DS was spat in the face, which was so outside of his frame of reference that he had no understanding of how bad it is and would never have told anyone) as much as you like, you can still be kicked under the table/have to endure deliberate meanness. All the lunchtime supervisors might do is insist you remain at your table rather than walk away from the situation, as you haven't eaten enough yet.
So I see many good reasons for delaying the start date, and/or going part time.
However every child is different, and though DS' 'full-time from the start' reception experience was pretty awful, we will still be sending DD from the start and will probably trial full time (the teacher has assured us that we could change back to part time if it wasn't working). Mainly because DD is such a different child than DS. I am hopeful that she will benefit academically from school where DS didn't; and that the social stuff will leave her untouched, whereas DS was hugely sensitive and aware of social relations, and therefore felt every rejection/isolation/put down very keenly.
So IMO it depends also on the particular child.