Lots of big companies/organisations have 'graduate schemes' that usually open so that students can start applying in their final year at university but are not just for final year students (in fact, I would go so far as to say that the majority of people who get on them have had at least a year out of univerisity). Yes, it is probably more unusual for people to be older but it's not an issue.
The idea is you train on the job and should progress relatively quickly.
I see that you do have a specific field in mind which would not lend itself to a grad scheme, but most major companies have them - supermarkets, banks, companies like Unilever... There are also public sector ones: Fast Stream for the Civil Service, Teach First as has been mentioned and a new-ish one for social workers I think. This sort of thing might be more appealing to you based on what you've said.
One issue you may find with grad schemes is that there are often 'postings' for 3/6/12 months so you may find an expectation to move around - not ideal with a family! This is something you would have to look into for individual schemes. I believe the public sector ones are better in terms of letting you stay in one place.
I had a career change at 26, wentback to university, and then it took me a year after graduating to find a job I wanted, starting at the bottom of the ladder and taking a pay cut from my previous career, so I don't think your age will be a problem, but you can't expect to catch up with your peers immediately.