I would absolutely consider Evanston (as well as Skokie) if the commute is to Lake Forest. Both places have rental options aplenty so you could get your bearings while looking for somewhere more permanent. Both also have a wide range of house prices if you decided to buy. Solid schools. Evanston has its own high school district while Skokie joins with Niles and Morton Grove, iirc.
Schaumburg would be quite a distance to cover, with the routes quite congested. I would knock it out of the equation if I were you. A commute from Deerfield to Lake Forest would be very workable.
In addition, the North Shore communities of Winnetka, Wilmette, Highland Park and Kenilworth might be worth looking into. They are very pricey though, and rentals might be hard to find, but schools are fantastic, and they lovely places to live. House proves are extremely high.
Or to begin with, the northern reaches of Chicago proper might work. Albany Park would get you onto I-94 north. Lincolnwood is also convenient to I-94.
Forest Glen/Sauganash/Edgebrook are three blended Chicago communities that are very nice, also close to I-94. Good schools, nice quality of life, city fun not too distant. Also nice is North Ravenswood.
As you may well be putting down roots, high school quality choice would be a factor in your decision about an ultimate home.
Within Chicago the selective magnet schools are top tire, nationally. Neighbourhood schools in prosperous Chicago neighbourhoods are also really good but you are still dealing with the vagaries of a massive school system that has periodic funding crises. Evanston is really, really good. New Trier, a massive high school district with multiple campuses fed by the North Shore suburbs of Wilmette, Kenilworth, Winnetka, Glencoe, most of Northfield, and parts of Glenview, is another top tier district. Highland Park, Deerfield, Bannockburn, Riverwoods and Highfield form their own HS district, which is really good.
If you decide to stay, or even stay a while and then return to the UK, it might be a good idea for the whole to get citizenship. You can be dual citizens (at least for now...) This would make university choice, loan entitlement, federal aid entitlement, and maybe even financial aid for individual universities more straightforward for your DCs.