My husband and children are American citizens. I would like to travel to the US with them. ESTA is denied and visit visa is still in process. My mother in law hasnt been well since last year. Can I travel with my British passport under visa waiver.
To travel on a visa waiver and arrive by air you need an ESTA. Without one you will not be allowed to board a plane to the US.
Technically you might be able to fly to Canada or Mexico and drive across the border to the US. However if your ESTA has been denied that means the US has reason why they might not want you in the country, or at least that you are ineligible for the VWP. When you get to the border the officer will know that you have been denied an ESTA. Depending on circumstances there is a high likelihood that you would be turned away at the border.
Unfortunately, this is likely to be a problem. Your husband could apply for a change of status after you are here. Thus this looks like an attempt to enter pretending to be a tourist but with the real purpose being immigration. If you're already here you're not thrown out during the application process, but if you're not here you have to wait through the process before you're allowed to come here. Even getting a visa in this situation can be hard.