Hi, bobnl - I think you will have to readjust your expectation on either commute time, number of bedrooms or price.
If you need to travel between London, Luton and Reading, that means you need to be looking in an arc roughly between the M1 and the M4. Anywhere within that area that is also 45 mins from central London is going to be very pricey.
Places like Watford, High Wycombe or Egham would fit your location criteria, but there are very few 3 bed houses in your budget, and frankly, they don't look great.
Personally, if you can't stretch your budget, and you definitely need three bedrooms, I think you would be better actually looking at either Reading (more expensive, nicer) or Luton (cheaper, not as nice). Both are commutable to London, although it is a bit of a trek between the two.
However - don't forget to calculate the cost of travel into your budget. A season pass might not be the best thing for you if you're not in London 5 days a week, but to give you an idea, they can run to several thousand pounds per year.