Often, there are midwives or healers that are introduced. These could be ancient vampires with special knowledge of hybrid births. Another common new character is a seer or prophet who can predict the future of Bella's baby. Sometimes, there are also new werewolves who come to protect Bella during the birth because they sense a great change is coming.
Often, there are new vampire characters. Maybe a vampire who was once a famous historical figure, like a medieval knight turned vampire. There could also be a witch - vampire hybrid who becomes Bella's friend or foe. Another common addition is a vampire with a special power related to the mind, who tries to influence Bella after her transformation.
The Cullen family usually plays a significant role. Carlisle, with his medical knowledge (even if it's vampire - related in the Twilight context), might be involved in ensuring a safe birth. Alice could be there, perhaps using her gift to foresee any potential problems during the birth. Even the other members of the Cullen family would be around to offer their support in different ways.
In certain fanfictions, a new hybrid character that is part - vampire, part - something else (maybe an ancient mythical creature) is introduced. This character challenges the existing norms of the Twilight universe. Also, a new human character who is immune to vampire compulsion in a different way than Bella was can be really intriguing. They might use this immunity to their advantage in various ways. And then there are characters who are descendants of long - forgotten vampire or werewolf lines, and their presence stirs up old rivalries and new alliances.
Well, a plot twist might be that the baby has a special connection to an ancient vampire clan that no one knew about. This could lead to a whole new storyline where they try to claim the baby. Also, Bella could experience a vision during the birth that reveals a future event that changes the course of the 'Twilight' world. Maybe she sees a great war coming and this knowledge makes everyone around her change their plans. It could also be that the baby's birth causes some sort of magical ripple effect that starts to change the abilities of other vampires in the area.
There could be other young witches or wizards who are also on the run from something or have their own issues with the wizarding establishment. They bond with Harry over their shared experiences and together they face new challenges. Maybe a young witch who has unique powers related to nature and she helps Harry understand the more elemental aspects of magic.
Often, a theme is the transformation of the relationship. The experience of giving birth can bring the characters closer or create new tensions. Also, the mystery of the birth itself. Since Twilight has its own set of supernatural rules, the birth process might deviate from normal human births. There could be elements of magic or special abilities involved in the pregnancy and birth. And of course, the protection of the unborn child, as there could be threats from other vampires or supernatural beings in the Twilight universe.
Bella's mother or a close female friend could also be main characters. Her mother could share her own experiences of giving birth and offer practical advice. The close female friend could be there to hold Bella's hand, offer comfort, and maybe even crack a few jokes to lighten the mood during the intense moments of the birth.
The Cullen family members are also key. Carlisle, as the patriarch of the vampire family, would be involved in the medical aspects of Bella's pregnancy and birth. Esme, with her motherly nature, would be there to support Bella. Alice, with her ability to see the future, could be crucial in predicting any problems during the birth. Jasper, too, would be part of the family's support system, even if his power of influencing emotions might not be directly related to the birth.
Bella is obviously one of the most popular. Since she is the main character in the original series, in fanfiction she often remains the central focus. Her relationships with the Cullens, especially Edward, are often explored in great depth.
The baby, of course. Growing up without a mother would be a huge part of the story. The father would also be deeply affected, having to raise the child alone and deal with the grief of losing his partner.