Our experience was that finding a full-time nursery school place for our pre-school son was no problem as long as we could find pounds 100 per week (almost the whole of my wife's salary initially, but a practical solution none the less).
What is virtually impossible is to find a school which offers pre-school and after-school arrangements that are compatible with both parents working full time, even where both employers are prepared to be very flexible. After much searching, the only schools we found which offered this provision were fee-paying; the experience of friends around the country confirms that this situation is widespread if not universal. An even greater problem is school holidays - in particular (for those able to use state schools) "Baker Days".
If we, as a professional two-parent family, find school-age child care a big practical and financial problem, then it is not surprising that most single parents view finding employment which will allow them to live from their own efforts rather than on state support as a practical impossibility.