If you were stopped by police on the road, the road into the airport or at the airport, do you really think emotional support is a valid exception (i.e. emergency) to justify breaking lockdown rules over any other UK citizen seeing their family? I really doubt it. During lockdown it doesn't matter whether family are 4,000 miles away or 30.
Wait until lockdown is over. Remove all risk. BA have a buy with confidence so you can receive a voucher until April 2022 if things don't work out or change your mind. Which you'll likely use with family in the US.
A last point. Cases are soaring as we're entering winter. Is it really worth the risk? My wife hasn't seen her parents in over 1.5 years. Was due to go over in July, cancelled. Then a week from now, cancelled (lockdown). Then we thought Christmas but we've decided against as the cases are just too high.