Greater London has almost no grazing within it, so buying "in London" really means buying at its edges — boroughs like Havering, Enfield, Hillingdon or Bromley still have working yards and some genuine countryside, but the overwhelming majority of London-based families end up looking out into Essex, Hertfordshire, Kent or Surrey for an actual pony to keep. That is completely normal, not a compromise.
If you are searching from London, the practical question is less "which pony" and more "which direction" — decide early whether you can realistically keep a pony on a yard within the M25 or whether you will be commuting out to a livery in a neighbouring county, since that shapes your whole search. Whichever you choose, factor daily travel time to reach the pony into the decision as seriously as the pony itself.