Update:
My wife received an offer for 3 points and a £100 fine for the careless driving offence. She called the number on the NIP and was told some very basic details although it didn't explain in full the reason why she was being prosecuted for the offence (i.e. there was no mention of the person stepping in to the road, only driving at excessive speed and it was mentioned in the statement that my wife made a gesture too). She didn't mention the gesture to me as she didn't think it was a big deal at the time, but it turned out to play an important part. Another option was for an officer to call her to talk through the allegation in more detail so she went for that option. My wife at this point had decided that having looked at the footage again that she'd take it on the chin and take the points and the fine.
We received a knock at the door over the weekend. It was the police, more specifically it was the individual in the dashcam footage. Obviously that explains that he was an off duty policeman at the time. We invited him in and he asked my wife if she knew why he was there.
Credit to my wife she said she was aware, she had reviewed the dashcam and concedes she was driving slightly faster than she should have been. Regarding the gesture, she was frustrated by him waving at her and shouldn't have made the gesture and was therefore sorry for doing so as it was unacceptable behavior. We had a bit of a laugh about it and how it was out of character for my wife.
The policeman explained she was actually lucky not to get charged with 2 offences: 1) driving without due care and attention as she took her hand off the wheel to make the gesture and 2) a public order offence for the gesture itself. The policeman said that even if the gesture hadn't been made, he thought she was driving too fast for the conditions and would have reported her anyway. He mentioned that he sees situations like this escalate out of control with road rage etc so feels this kind of education is important. Can't argue with that.
The policeman was very reasonable and said that because she admitted she was in the wrong and showed remorse he would drop the charge against her and to consider it a telling off and a warning. If she had played dumb or tried to justified her actions etc then he would not have dropped the charge.
I thought that was an incredibly effective way to deal with the situation. My wife said that even though she was relieved not to receive the points, it was more of a lesson/telling off as it was more personal. It also taught our son who was in the room at the time that honesty is the best policy and the best way to deal with situations like that. Sometimes these lessons are best coming from someone else so they actually take some notice.
A good outcome to a situation that probably should have been avoided. I'll have this in the back of my mind next time too