I can help you with some of this perhaps :-)
First of all, I'd recommend you read Helen Russell's "The Year of Living Danishly" - it's an easy read, quite funny, and gives some great information as to what is different in DK for British people. Interview with her here:
www.visitdenmark.co.uk/en-gb/denmark/british-expat-helen-russell-living-life-danish-way
Another good book is How to be Danish by Patrick Kingsley. Read both!
Childcare - I'm guessing you'd be looking for an international nursery (vuggestue), I don't know about them, but 'normal Danish ones' are available for children from 6 months.
Cost of renting - it's hard finding good flats in Copenhagen, but with a good budget (ie a UK budget ;-) ) you should be fine. There are rarely fixed terms - normally rental contracts run "forever" with a 1-3 month notice period. This could be different with flats rented via agencies/by an employer. Quite often you have to give a deposit too. Costs vary so much, but sounds like you want a "nice big" place. Found this example: 123 square metres for approx 20 000 DKK per month (very expensive in Danish terms - and very big) www.akutbolig.dk/vis/239691
Standards of living are much higher than in the UK. Flats, or any homes, are warm heating is not as extortionate as in the UK, since the building standards here are/have been as crap. Many "old" flats have quite small kitchens, and a separate dining room/eating area, but many have also been redone to have a "kitchen diner".
Healthcare - assuming you'd "become Danish" you register with a GP and it's normally not that hard to get an appointment. Kind of depends on the area though - I hear masses of people complaining a lot about it in the UK, but I've never had any problems in my surgery.
Healthcare is free, but dentists are expensive. Sweden's dentists are much cheaper so a fair number of people travel there to see a dentist, it's cheaper and the Swedish krona has fallen so much, and the Danish one is "too strong". (Some people do a lot of shopping in Sweden too!)
Taxes - this is a minefield and I'd advise you to get separate advice on this. There is a scheme for 'foreign experts' - if your new company can manage to get you under that, you'd pay much less. It's only available for a few years, so basically a way of attracting foreign expertise that would otherwise not want to work there.
Language - everyone speaks at least basic English, normally good English. Very often English speakers complain that they don't get to learn the languages since everybody just speaks English to them.
Many, many people don't have cars, especially if you live in the "centre". Everybody cycles - there are cycle jams on the main routes into town in the morning. Children are transported in cargo bikes like these:www.christianiabikes.com/
It's very expensive to have a car (taxes) so that's one reason, but you really don't need one either. Good trains, tubes and buses.
To be honest, I don't find the dark that much different between CPH and London. There is of course some difference, but nothing I think about really.
Costs of services - definitely much more expensive than in the UK (since everyone gets a living wage), but other things are cheaper, public transport for example.
Work/life balance - definitely better than in the UK. You'd probably work more than public sector Danes, but still very different to the UK I'd say. Helen Russell goes into some detail on this, being super surprised one Friday when her partner gets home from work at 3 or something, and they learn that that's normal in DK to get family time!
Hope this helped a bit. Good luck with the move - I don't think you'll regret it. There are loads of expats in CPH too that will be happy to help, and welcome you!