Fully dynamic algorithms for strong components of a directed graph
I know it is an older question but this research paper has a relatively straight forward O(m^(3/2)) algorithm for incrementally maintaining them: https://arxiv.org/pdf/1105.2397.pdf
Here is one with O(m) algorithm for decrementally maintaining them: https://arxiv.org/pdf/1901.03615.pdf (from 2019 and it improves on the O(m*sqrt(n)) algorithm of https://u.cs.biu.ac.il/~liamr/scc-new.pdf)
I am not sure there is anything better for doing both operations.  Part of the issue is the data structure used for maintaining them.  Incremental algorithms are typically using Tarjan's Union-Find for fast partitioning of disjoint sets.  While decremental algorithms are using an Even-Shiloach(ES)-tree, as the breadth-first-search representation matches the problem.
As for a unified data structure which efficiently can be used for a fully online SCC maintenance, that would be progress we hope to see soon.