Wouldn't a fiancee visa be more suitable in this case? You'd have six months in which to live together, and if things worked out, you could marry and remain in the UK without uprooting your children needlessly back and forth. If things didn't work out, you could return to the US. There have been several people who have moved to the UK on fiancee visas and for whatever reasons things have not worked out and they've moved back, so you'd always have that option.
Fiance / Fiancee Visa Requirements
The sponsor and the applicant must intend to get married within a six month period.
They must have met each other.
They must intend to live together permanently.
The applicant must be able to support him/herself, or be supported by their partner, without access to public funds.
There must be adequate accommodation in place.
Conditions Of the Visa
Successful applicants are granted a 6 month visa in which to come to the UK and get married. They are not allowed to work while on this visa.
Applicants must then apply for a marriage visa once married, which if successful, gives them 2 years to live and work without restrictions in the UK.
If the relationship is still continuing after this 2 year period, applicants may apply for Indefinite Leave to Remain (ILR).