Namechanged as this is identifying.
I live in the region and have friends with children at two of the schools you mentioned, plus acquaintances with kids at the third.
Englefield Green Infant school has in recent years been slightly undersubscribed, so you have good chances of getting in no matter where you live. Similar for Christchurch CofE - due to low numbers of applications they have recently reduced their PAN from 40 to 30 and now run only three classes (no longer 4 mixed-age classes). None of the three years is currently full. Again that means you would get in no matter where you live. In fact, Christchurch is one of those schools that gets allocated to people who didn't get any of their preferences.
For both schools that means you would be likely to have a small class. My friends' kids have classes of 25 or 26 or thereabouts.
As Christchurch is single-form entry infant school and classes aren't full, it is a very small school with less than 90 children. My friends say that the head teacher is very friendly with the children, but strict with the parents! Children will go up and give her a hug. My friends are generally happy there, but not everyone is - some people from their classes have moved to different schools as places became available, e.g. to Trumps Green infant school. Things can change quickly with new teachers/new policies, but my friends reported that they had spelling lists to learn from reception, and quite a lot of homework from reception as well. The kind of homework that needs lots of parental help.
It is a church school where religion is important. It has an associated church and school has expectations that enrolled children will go to mass at that church (rather than any they may have previously been attending, or none); especially for (regular!) special masses.
It is in a beautiful setting including a little bit of forest on the grounds, which I think they use well for outdoor learning.
As it is never full, a good number of children who attend the school live quite far away (e.g. 20 minutes by car).
My friends have mentioned concerns that if the school continues to shrink, it may soon no longer be viable and have to close. But that may be completely unfounded.
As it is an infant school, you would have to consider where to go on from there. I believe most children go either to St.Jude's CofE or St.Ann's Heath. I think (but may be wrong) that the main secondary school for those Junior schools is Magna Carta.
You will know Englefield Green Infant school already as your child attends nursery there. Obviously it is a lot bigger than Christchurch, with 6 classes plus the nursery kids, so around 200 children. As it tends not to be full, you would have good chances getting in. However, again you need to consider what happens next. Nearly all children go on to St.Judes CofE - however I don't think there is an automatic preference (check that! in St.Judes admissions criteria). That would mean that there is a risk that your child is the one who doesn't get a place in the Junior school that all their friends go to. Perhaps leaving you struggling to find a Junior school you like.
There are always children who move away from Englefield Green Infants, e.g. to Manorcroft; but there are also always children who transfer INTO it from e.g. Manorcroft. Compared to Manorcroft which is said to be quite strict, with homework and pressure, Englefield Green Infants is said to be very play-based (not just in reception).
Thorpe CofE I believe is always oversubscribed, it is a very popular school. You would only get in if you are in a high admissions category, or if your child has a statement naming the school. The school is known to be excellent for children with SEN. Everyone I know who has children there (SEN or not) is extraordinarily happy with the school.
There are a lot of other schools in the area between these three! E.g. Manorcroft, Trumps Green, Pooley Green, Hythe, ... and depending where you live, you would have chances getting into any of them. Trumps Green admitted up to over 3 miles distance recently (and before that, was not full, as they had started a second form intake), and Hythe School which used to have a small catchment, has recently started a second form (so takes 60 rather than 30) and hasn't been full since.
You need to decide what is important to you. For me, distance was an important factor (walkable distance/need to drive). Size of school. The role of religion. Strict with lots of homework, or more play based, relaxed. Infant school or through primary. What schools they tend to feed into. Perhaps even school hours - some schools have days that are 45 minutes longer than other schools. If you add a long journey to get there, that can mean a big difference every day. E.g. 9-3 school around the corner, you leave the house at 8:50 and get home at 3:10. Meaning you get up at reasonable time, and leaving lots of time for play/after school activities/chilling (or, causing lots of childcare headaches). Compared to 8:45-3:30 school half an hour away. You leave home at 8:10 and get home at past 4pm. Leaving you struggling to get any homework or playtime or activities done before tea and bedtime, and struggling to get child up and out in the morning.
Good luck with deciding!