Am in the "it's way too soon for the DC to meet their parent's new gf / bf" camp. A matter of months is not enough for bereaved DC, including teens, to have to deal with a parent's new gf/bf IMO.
Also think it'd be v hard to be a single parent, maybe working, and have time/energy for a new relationship and DC.
It's obviously important whether the new person is good to the DC, and this might be hard to work out while in the "honeymoon" phase of a new relationship, and without introducing them to the DC.
Have recently discussed this with DH, he has direct experience from the DCs' point of view from his teens. We (it's easy hypothetically!) felt it would be disrespectful to the deceased one and unfair to our bereaved DC for the one who was left to "move on" too soon, but whenever a relationship and introduction to the DC did happen, if the person wasn't good to the DC then the DC should come first. If am being honest though I think my DH would be more likely than me to find someone quickly.
My grandmother was widowed suddenly in the 1950s, she was young and had two DC aged 4 and 18 months, no compensation, life insurance, welfare etc, she felt her only real option was marry again quickly, unfortunately she married a horrible, physically and emotionally abusive (to her and the DC) person 