Winchester, St. Paul's, or Eton?

@nolascrack well with Eton taking the lion's share (god knows for what good and bad reasons), difficult for any school to post even one PM. Other forces may be in play perhaps?

It looks like you can be groomed to be a PM if you are decently smart (as all were, and some very very smart indeed to be fair) but not way you can pull it off if you don't have what it takes to be a Chancellor. Thanks god some things in this world are more than merit than privilege.

You have been let down by current school for not giving you heads up about this much earlier so you could properly research

Or to put it another way Winchester has never produced a single Prime Minster..and St Pauls just the one..

SPS, Winchester and Westminster are eminently more intellectual and STEM than Eton. Perhaps less so in recent years but the profile is still very privileged compared to the other top schools.
A bit of trivia: no Chancellor of the Exchequer came from Eton at least in the past 40 years (haven't checked before) but produced plenty of well oiled and versed PMs. Chancellors have come from everywhere in the country, from grammar to normal state schools, and also including from SPS, Westminster and WinColl (current one). That tells you the profile a bit.... just saying.
waits for Pete to jump at this with plenty of other trivia to refute.

I'm curious - did your son only start at the prep school this year?

Because no headmaster of an academic prep school would be beginning discussions about senior school in the middle of yr 6. They would have missed the deadlines for half the obvious places.

Presumably you have already made applications elsewhere and have some indication of offers or waiting list places?

Rather a shame you didn't begin investigating senior school entrance before this. As pps have said, you've missed the normal entrance process for at least two of those schools - leaving you with no option but the yr 8 scholarship route. This is far from ideal - but not impossible.

Have you made it clear to your current headmaster that you only want a traditional boarding school? And do they have a record of scholarship success at the two mentioned?

You need to contact the schools as soon as you can so as to be fully informed of their current scholarship application processes. And you will definitely need to ensure you have back-up offers from other schools. You cannot rely on being offered one of their extremely scarce scholarships.

(Having said all that, who knows how public school places will be affected by the new economic climate?)

bullying and academic quality?

These are vague red herrings. If you're already at a prep school you should already know all you need to know about the relative merits of the public schools in question.

The OP is talking about sitting the scholarship exams I think you can sit Election without sitting the ISEB pre test you would have to speak to the admissions dept to clarify this I dint know about the others. Competition for a scholarship at all three is very fierce and its not a done deal however bright your DS is you will need a fall back. Does your current school have a history of preparing a boy for scholarships into any of them this will definitely help. Some preps have a proven track record. I suspect most getting scholarships into SPS come from SPS junior or whatever its called now (used to be Collet Court).
All three excellent and different schools, some boys will suit all three but some will suit just one. A bright boy is likely to do will do well at all of them. Once they start having open days again go and have look and speak to the admissions dept they will happily tell you what they are looking for and provide past papers etc.

SPS has few boarders and I far as I'm aware mainly 6th form boarders.
Academically there is very little to choose from the three if you're looking at results although I suspect SPS has the slight edge. My DS went to Winchester (different head) it was unashamedly intellectual (other parents with sons at both Eton and win Coll said this) lots of intellectual activities, no compulsory drama in the curriculum although loads of extra curricular drama, sport was not big many managed to do no sport at all after the first few terms, it was liberal, little uniform, little meaningless ritual, a good place for lone wolfs, eccentrics and those who value there individuality as this is ever much encouraged hands off parenting (and thankfully few compulsory social events). Boys weren't allowed personal computers for the first two years everything is written by hand unless you have dyslexia etc. HM's are very significant in a boys school life. Its in my opinion living in a bit of a time warp especially in comparison with SPS which is on the other end of the scale; modern a very 21st century school outward (you cant help but be impressed when you go round) and we spent over a year choosing between the two but eventually we felt Winchester would suit my DS better it has a unique tranquility about it (I've worked in other very well known boarding schools so am speaking form experience).

Having said all of this this it has a new head so I cant comment on it now.
There is bullying at every school in every sector my DS said physical bullying was very much frowned upon by the boys but most boys are highly articulate so its inevitable that there will be some verbal bullying. I have no experience of how the school response to bullying but we found it a very caring environment so can't imagine its tolerated.
Good luck.

Eton

Also too late for Aston

Think you are too late for Winch, they sat the iseb common pre test in autumn year six, interviewed done last term and places offered soon for year nine entry 2022.

St Paul's is very much a London day school with a few boarders. He definitely wouldn't get a typical boarding school education there.
Has he done any pre-tests? I may be wrong but I thought Eton and Winchester both did pre-tests in Year 6.