Any excuse to look at houses on Rightmove 
All of these are within 30 miles of Leighton Buzzard and priced at £1.25k. All have 5+ bedrooms and are inside the admission area (and within a mile or less) of an Ofsted Good or Outstanding school that wasn't oversubscribed in 2016:
www.rightmove.co.uk/property-for-sale/property-63486395.html
Hitchen, Mary Exton Primary School
www.rightmove.co.uk/property-for-sale/property-42058677.html
London, West Grove Primary School
www.rightmove.co.uk/property-for-sale/property-63427355.html
Great Kingshill, Widmer End Combined School
www.rightmove.co.uk/property-for-sale/property-63721073.html
London N3, Tudor Primary School
www.rightmove.co.uk/property-for-sale/property-44433498.html
Northampton, Wootton Primary School
www.rightmove.co.uk/property-for-sale/property-57499924.html
Chalfont St Peter, Bucks, Robertswood School
www.rightmove.co.uk/property-for-sale/property-46167783.html
Princes Risborough, Great Kimble CofE School
www.rightmove.co.uk/property-for-sale/property-62771456.html
Stanmore, St John's CofE School
www.rightmove.co.uk/property-for-sale/property-43997235.html
Biddenham, Westfield School
www.rightmove.co.uk/property-for-sale/property-53770897.html
London W7, St Mark's Primary School, St Joseph's Catholic Primary School, Mayfield Primary School, Dormers Wells Infant School all within a mile.
www.rightmove.co.uk/property-for-sale/property-63123425.html
Eastcote HA4, Coteford Infant School
www.rightmove.co.uk/property-for-sale/property-38633457.html
Oxford, East Oxford Primary School, New Hinksey CofE Primary School
That was just the first two pages of search results. If you're happy to travel more than a mile to a school then there are more that meet your criteria. I don't personally know anything about the transport links to London or these particular schools, perhaps others do.
I have major house envy now after looking at all those beautiful houses!