Do you mind me asking if your older child is your DD or DS? Is the older one going into year 5, 6 or 7? And if the younger one (at 8) would be going into year 3 or 4? It really does change the options!
I think Hanford would be your best bet for your DD - I would worry less about the formality and more about the environment, ethos and community of the school. As much as I love the structure and 'formality' of boarding schools, there will be plenty of that at senior school. Hanford seems to tick all your boxes - especially ponies! Having said that, Godstowe is the girls prep so if you could be sure that there would be a few other junior boarders in her year I would seriously consider it.
But for a girl many families wait until year 7 for boarding and then go for the big year 7-13 girls boarding schools. Alternatively there are some girls schools that have prep and junior schools attached and boarders can start in junior but usually year 4 or 5 not year 3 (e.g. Godolphin (Salisbury), Badminton Junior, Thornton College, Adcote School or Royal Masonic School for Girls. Also Moreton Hall Prep starts boarding at year 3 and is co-ed to end of year 6, then girls only from year 7 in the senior school). These don't necessarily fit your geographical preferences just putting the idea out there.
Also note that the number of boarders usually increases year-on-year in these types of schools, number can be very low at the junior end, especially at weekends. I have seen this work well, one school I worked at had a junior boarder in two-three nights a week who was three years younger than the next boarder (she was only six!) and she loved it - she was like the little sister to seventy girls and she stayed on and became a full boarder end of year 6. I have also seen it not work - with one or two little girls alone in junior boarding houses and unhappy about it - for junior boarding I would be asking about weekend numbers very carefully.
Alternatively you could put your daughter in a co-ed prep and move her to a girls school at year 7. If your daughter is the younger child, this might line up with your children changing schools at the same time? There are an absolutely huge number of co-ed preps - I probably haven't even heard of half of them, and many offer boarding in some form or another. I have heard good things about Windlesham House, Hazelgrove, and Highfield and Brookham (the latter (H&B) is one school) for full boarding. I have also heard pretty good things about Vinehall, Sandroyd and Brambletye but more through the grapevine than from people directly involved if you see what I mean.
Not sure re geography again, but for your son have you looked at Sunningdale and Elstree? Boarding has become so popular at Sunningdale I believe they are phasing out day places to just be a boarding prep again. I have seen summer fields has already been recommended too.
Ludgrove is great in that all the boys are boarders but I believe they are fortnightly - i.e. go home every second weekend. This may be tricky if you are overseas though I am sure they have some type of arrangement for this.
With your children it can sometimes feel like you have limited options - e.g. looking for junior all girls boarding for your DD. Don't fall into the trap of thinking this way. Junior boarding is a "buyer's market" as there are more options than children for all but the most exclusive schools. Finding the right school for each child is more important. You want to make sure that the child fits in well and loves it. Most schools will bite your arm off for a junior full boarder - make sure the school is right for your child.
A top tip is to look for a prep with a large number of military families. I don't mean a military boarding school, I mean one where soldiers send their kids on continuing education allowances (CEA). They will likely be full boarding pupils who won't revert to groups speaking a language your DC can't when unsupervised. You may be a military family yourselves - I don't know!
Alternatively look for school with compulsory year 7/8 boarding. This will ensure plenty of company in the senior years.
Look at the things that will really matter: pastoral care and safeguarding would be number 1. Really drill down on this - different schools have different policies and you may be surprised what bothers you. For example I have worked in a girls school where men were not allowed in boarding - for example both the matron and a gap student had to supervise the plumber who came to fix a shower. At the other end I have seen schools where the head of junior boarding was a man who had unsupervised care over young girls.
Speak to any parents you can. Leavers' destinations are also very important - you want broad opportunities, year 9 will come faster than you know. Make sure you know the "pressure level" at the school. I have seen schools that publish weekly or fortnightly rankings of the class - alternately I have seen schools that won't even host an annual prize giving. And if your child has one or two hobbies they love, please pick a school that offers them - it's not like a day school where you can make it up in the evenings.
Check how the school functions - are girls and boys kept separate at co-ed preps? Are the sports options different - no point sending a cricket mad DD to a school where only boys are allowed on the team. No point sending your kids to a co-ed school to support each other if they are in separate houses and only see each from a distance. Also check on some things e.g. I have seen schools where the choir is almost entirely girls and this has put talented boys off. They may be better off in a school with less of a choir provision but a better boy/girl mix or boys only. Doesn't matter how good the choir is if your kid won't participate.
I would also say make back up plans. If your child comes home and says "Mum, I can't do this, I thought I was ready but I'm not and I don't want to go back." You need to have something in place. Reality can be quite different to what has been imagined. For your DD and Hanford, I believe they offer "one-term" places (or at least did before covid). Maybe your DD could try a term and if she isn't ready could come home again?
Not sure of your situation and what is possible for you. I have seen everything from international boarders who started at 8 and never left the UK again (parent was a judge in unstable developing nation with credible death threats issued against the children) right through to kids full boarding in the same village they lived in - by choice!
Don't be afraid if the school is not working out but they really can't come home to look at switching schools. I would say if something is terribly wrong take them out immediately, otherwise give it a full term at least but if it's been weeks and weeks and they haven't settled - have back up plans.
Good luck.