I did a PGCE at Kings College, London around the turn of the millennium to train to be a teacher. My personal tutor there was on occasion an advisor to the government and through chats with her I found the main problem from their perspective was that (a) the government just ignored their advice and (b) when they didn't it was generally some idea of 'best practice' which the government would try and implement on a national basis without taking into account that different approaches might be better for different regions/schools/individuals. So my approach would definitely include listening to the experts and also having some element of flexibility.
Being top of the class at pedagogy (the theory) wasn't enough to stop me failing in classroom management (and you have to pass every element to pass); but I retried a few years later in Derbyshire - with the same results. Which increased my theoretical knowledge even further but without any useful outcome. It also means I've had experience of teaching in 4 very different schools: a private school (rich, less pupils, but also some very intelligent approaches to learning missing from the State sector); an inner city school (poor neighbourhood but rich school because of subsidies for poor neighbourhoods, very low standards from the intake area); a good state secondary school (poor catchment area but really good teachers and school set up) and a ghastly mediocre state school (poor catchment area but generally awful teaching staff and a system set up for babysitting rather than education).
My first thought would be that the fundamental aim of the education department was to improve education - any ideas about social engineering would be a 'bonus' if they could be implemented without damaging the core objective of learning.
For primary schools I'd make all the pupil assessment internal and teacher based. I'd have a review of best practice teaching methods and encourage schools to follow this, but allow the flexibility not to do so when teaching staff don't think it's best for their individual situations. I would add regular inspections to give some basis of comparison for parents in the neighbourhood to compare schools and also to check whether any deviation from the best practice methods weren't harming pupils chances. Low scores might result in an encouragement to change their ways, but failed inspections could result in Ofsted (or it's successor) imposing a change in their practices.
At the beginning of year 7 I'd have a standardised test for all secondary school entrants so the school can not only assess the correct sets for each subject that pupils need to go into - but also a baseline so that the GCSE results for that year will show the added value score each school has made from the intake's initial potential. I'd get rid of streaming if any school still does it, but introduce setting if they didn't have that.
I'd introduce touch typing courses (within PSHE/citizenship - possibly even English, maybe?), because it's just such a useful skill for almost everyone to have. I think I'd approach term time holidays as - pupils would be allowed up to 7 days off during term time at the Head Teachers discretion (not including unavoidable things like doctors appointments etc.). Most term time holidays aren't in the slightest bit educational and if it means families have to go to the UK coast for the holiday because they can't afford to fly abroad then it's not really that big a deal. But if a family did have something which was actually meaningfully useful then it could be allowed as long as they take into account the advice of when it would be least disruptive to their child's education.
I'd make education mandatory up to 18, but if pupils failed their GCSE maths I'd introduce a supplementary Numeracy certificate for them to take for 2 years which would concentrate purely on the most useful arithmetic that people need for everyday life. They could also do vocational courses/apprenticeships alongside this unless these course included the Numeracy certificate within their curriculum.
I'd definitely ask for a departmental reorganisation so that Universities came under the remit of the Education Department rather than Business (as it is now). And I'd abandon the idea that having everyone stay on to higher education is a fundamental aim. I'd want regular inspection of universities - particularly looking at quality of teaching and drop out rates. If a university failed in either of these they could be downgraded to a higher education college (if they were primarily academic in nature) or polytechnic (if they were primarily vocational or technical in nature) - funding would obviously be impacted by such a downgrade.
I'd also set up an enquiry to try and improve the teaching profession. The problem at the moment is that a lot of schools would be improved by just having more teachers, but if you pushed that through then quality would fall. One approach would be to shove more money into pay - but that wouldn't necessarily improve the prestige of teaching until it got vast swathes of highly qualified graduates to choose teaching rather than alternatives and might not be the best approach anyway. It's a difficult one to resolve - hence the enquiry.
Other ideas would probably come as a result of expert consultation - but that'd be a start