My children are in Pilgrims Pre Prep and I have viewed Bedford Modern but my children aren't yet of an age to move up, so the other opinions I share will be my perspective based on experience and discussion with other parents who do have children there!
Pilgrims is a wonderful pre-prep. The opportunities for the children are fantastic, the style of teaching is wonderful. There has been a lot of turnover in their early years teaching staff lately but I feel it's a short term issue. Each term is based around a topic and the children really engage with this. However because they are prepping children for the 7+, you should be aware that it is demanding academically and this filters into the amount of homework/home support that you need to provide. The headmistress is really inspirational.
Challenges I have found have been mostly related to being a working parent: breakfast club doesn't open until 8am unlike many other schools. This means that it's impossible to drop your child off and get a train from the station as the car park is full. Some parents have got around this by paying for a premium parking spot at the station.
The school is lucky enough to have a car park, but it is extremely difficult to park in, unless you get to school to pick up 30+ minutes before collection time, especially where events are happening at the school.
The other irksome thing for me is that all after school clubs (one of the things that attracted me to the school) in Years 1 and 2 are run from 4-5pm, if not later. It makes for a very long day and has put me off placing my child in them, so they're not actually getting to make the most of the opportunities that are provided.
Communication between parents and teachers is excellent though, and the amount of support provided to my children has been great.
One of the reasons we are not continuing with the Harpur Trust into prep is the provision for girls though. I have yet to hear a good word (even from current parents) about the Girls School; I feel it suits a particular type of girl. Very assertive, shall we say. The campus isn't great also and I've heard a lot of parents complaining about drop off and pick up, as there's no clear provision given its location in the town centre.
With regards to Bedford Modern, it gets good results, it has lots of happy parents. It simply wasn't my idea of what I want an independent school to be.
Please also consider your commute to/from school/work - although Bedford isn't big, the journey times can be long due to traffic (and continuous ongoing roadworks). You might also consider if your child will take the school bus and what those routes are.
I'd strongly recommend you visit all the schools as soon as you can to get a feel for them.
However, I think you should keep an open mind with regards to the other independent schools in Bedford (Polam, Rushmere & St Andrews who are merging and will be building a joint site shortly).
Kimbolton School goes from 4-18 and is 15 minutes north of Bedford with a good bus service. Their entry in the Good School Guide recently is telling, and it has a completely different feel to the Harpur Trust schools.