How long ago did the old civilization go away? How many people are still around? Are there many remains of the old civilization? Did the former civilization prepare for this event and tried to leave seeds for a new civilization? If any, which ones?
A lot is said about minimum viable population, genetic variability, and inbreeding but these are not absolute and insurmountable problems. You will have more or less challenges based on your two initial people, and it may not be possible with any two. Pre-screening of the individuals could rule out many of the nastiest recessive genetic diseases. An alternative and less savory solution is controlling for genetic disease with basic knowledge of Mendelian inheritance and eugenics.
As a fun side note, the natives of the island Pingelap are an interesting example of a population that successfully rebuilt itself after a bottleneck of about 20 people. As a result, recessive color blindness is common, but their civilization was rebuilt.
https://en.wikipedia.org/wiki/Pingelap#Total_color_blindness
With respect to the time required to rebuild civilization, assuming no environmental constraints and the available knowledge for the past civilization, the primary limitation is the economic productivity of the population. If we ballpark the minimum population to produce modern technology at ~1 billion people, this would take 20 generations at 4 surviving children per woman. This is about 500 years at 25yrs per generation.
This seems fast but might not be crazy if you compare it to the to the state of humans and medieval technology in 1500 AD, and the rate of development since then.
https://www.britannica.com/science/minimum-viable-population...
If people were attempting interplanetary/interstellar colonization or planning for a serious interruption they could possibly stockpile millions of human eggs.
You need at least around a thousand people so that you avoid too close family relationships over the generations. Enough women to sustain birthrate over the 2.10 per woman number necessary for population stability. Even more births are necessary in a pre-modern society.
At least 90% of your workforce will be involved in obtaining food. The number might be lower under favorable conditions, but have that in mind as a rule of thumb. If your seed population has access to large domesticated animals or source of fish, it will help them concentrate on the main job, a.k.a rebulding civilization.
So, the rest of your population will be craftsmen, scientists, industry workers, and administrators that supposedly will facilitate the work of the others. Under administrators we can talk either bureaucrats or traders or whatever people you need for your distribution model. Craftsmen are people who build or obtain stuff that a single person cannot do on their own because it is time-consuming or requires specialized knowledge. This includes carpenters, smiths, farmacists, and so on.
If we take ancient cities as an example of relatively independent units that are able to survive more or less on their own, and if we calculate the surrounding villages as food base for the city, with approximate population of a city around 4-5 thousand and 5-10 times more in the villages, the figure will be around 40000-50000 people.
This culture must be well-organized, have written tradition, relatively free transfer of knowledge, stable environment, and space to expand. The future generations should be able to use the preserved knowledge for obtaining even more resources and have accelerated expansion. There is a tradeoff here, because the more knowledge you want preserved, the more people and resources should be involved in preserving it, which means more people needed to supply them with food, shelter, and parchment. So, try to keep only the knowledge necessary to build all the rest. Maybe good candidates are: math, logic, scientific method, mechanics, basic chemistry, biology, and astronomy (always good to know what part of the year it is if you want good agriculture).
These are all back of the envelope thoughts so might not hold out under careful examination.
>genetic mutation which took place 6-10,000 years ago and is the cause of the eye colour of all blue-eyed humans alive on the planet today
Having said that, ideal would not so much how many people but how diverse of a gene pool can you get?
Then add in human intervention ie. CRISPR if possible to then do gentle mutations to ensure the diversity keeps growing instead of shrinking once procreations starts.
My estimation: basically we need six to ten million individuals and 200 years.
P.S. for science fiction lovers: Antarctic would fit perfectly if the Earth axis shifts and the continent is placed in the temperate climate again but more time will be necessary to allow for the growth of forests first.