It's a mix. The show is based on historical events which are facts. For example, the Viking invasions and the existence of various kingdoms in England during that era are real. However, some of the specific characters' storylines and relationships are fictional. They are created to make the story more engaging and dramatic.
Uhtred is often a very popular character in fanfiction. His complex character, being a Saxon raised by Danes, offers a lot of room for different storylines.
There could be many. For example, some fanfictions might focus on alternate endings for certain characters. Maybe a character who died in the original story gets a new lease on life and goes on to achieve great things.
One common theme is the clash of cultures. Since The Last Kingdom focuses on Saxons and Vikings focuses on, well, Vikings, when they cross over, there's always a lot of exploration of how these different cultures interact. Another theme is the idea of loyalty. Characters from both shows have strong loyalties to their own groups, and in the crossovers, this loyalty is often tested.
Another could be an OC who is a foreigner trying to make a place for themselves in the Saxon - Danish world of 'The Last Kingdom'. They might face discrimination but also bring new ideas and skills, like a different style of combat or farming. For example, they could introduce a new type of crop that could thrive in the local climate and change the food situation in a small village.
Brida is also a very popular choice. Her strong - willed and sometimes vengeful nature makes for exciting storylines. Fanfiction can delve deeper into her past and what led her to become the way she is.