I like the idea of drama/ acting classes. It will teach them so much about their own voices and give them confidence to stand and speak in front of their classmates and an audience. A choir is also a good idea.
Education - what schools are they currently in? Would they get into a competitive grammar school? Regardless, make sure they work to the best standard they're capable of, and encourage them to take their school performance seriously.
Sport - is there a tennis club or cricket or hockey club they could join? Any opportunities to participate in a more obscure sport like fencing?
Reading! Focus on British 20th century classics, especially anything with a hint of humour. That's advice for you and your H.
PG Wodehouse, Evelyn Waugh both wrote some funny books. Incorporate vocabulary and turns of phrase from books you read into your everyday speech.
Encourage the children to read too, obv. Chat with them about what they're reading. Push them to expand their taste and improve their level, but don't pooh pooh their choices. Your aim is to build their confidence, not make them feel 'less than'.
Don't fret about a certain level of bad behaviour or loudness on the part of the children. Respond as if you own whatever place you're in when they start being naughty. Don't act embarrassed. Don't dress up too much when you're going places.
Talk to the children, ask them their views, help them talk about their feelings, make them feel they're important. Do family fun - a movie night, skating together, long walks, video games.
Make sure their friends are welcome in your home, and talk to the friends too.
Don't pass on your own insecurities. Appreciate and pass on the great parts of your upbringing and culture. Make sure your children are proud of you and their wider family.
You and your H could join local groups and get used to dealing with all sorts of other people.
You could also volunteer somewhere as a family, and seek out ways you and the children can be leaders (organise a pick up rubbish day, organise a food bank collection, etc).
This might be controversial, but joining your local CoE congregation might be a good way to introduce the family to a wider social circle. It could also offer opportunities for volunteering.