This will end badly
A number of details are omitted so I've filled in with some assumptions.
Number of children annually: from the comments, we start with 12/yr but grow to "a few dozen a year" by year 50. Since "a few" isn't really something I can plug into excel, I'm going to assume few = 3.5, so:
- year 1 = 12 children/yr
- year 50 = 42 children/yr
For simplicity, I assumed this is a simple linear growth rate.
What about grandchildren? I'm assuming children start having children at 20 (this isn't really accurate for this type of society but it's a safe, conservative guess). In this type of society, we assume a very high childhood death rate. But, since these are super-children, I'm going to 70% of children make it to adulthood. I's also going to assume their reproductive rates are normal for their society (this also isn't very realistic but, again, I'm being conservative). That means that each child has about a 1/3 chance of having a child each year.
Great-grandchildren? Same as with grandchildren but I'm going to reduce number making adulthood to 50%.
We're also told that the tribe he joins starts with about 100 people but that other tribes are folded in over time. How many and how big isn't stated. I'm going to assume a total of 5 tribes with 100 people each make up his larger community after 50 years. To make the math simpler, I'm going to assume they all fold in the first year. Societies in this state usually have very low (near zero) growth rates. In the name of keeping my estimate conservative, I'm actually going to assume that these 500 people have a growth rate of 1% annually. This is near modern levels but they're benefiting from whatever tech he has.
working out the numbers
After 50 years, our time traveler will have 1350 children. Some will have died and some will still be children but using my assumptions above, he'll have 1000 living children including 460 adult children.
Together, those 460 adult children would have produced 2100 grandchildren. Again, some will have died and some will still be children but by year 50 there should be 1325 grandchildren including 110 adult grandchildren.
Those 110 grandchildren will have produced 155 great-grandchildren, all of which will still by too young to reproduce.
Total descendants at year 50:
- children: 1000
- grandchildren: 1325
- great-grandchildren: 155
- TOTAL: 2480
By comparison, the "normal" people in this super-tribe would have increased to about 825.
- Time travel descendants: 2480
- Normal population: 825
- TOTAL: 3305
The most prolific "super-breeder" known in history is Genghis Khan whose descendants represent 8% of the Mongolian population (where they're concentrated) after several centuries. This time traveler's descendants are at 75% of the local population after only 50 years.
That's going to leave a mark
but wait, there's more
The question also states that the nanomachines will cause "a very real uptick in health for his children as a result of the nanomachines working to a lesser degree for them due to shared genetic code". This suggests they both proliferate across persons and that they are tied to the time traveler's DNA.
That would suggest the introduction of an artificial evolutionary pressure. There is an evolutionary benefit to having genes similar to the time traveler. Beyond simple descent, you now have the random mutations of evolution selecting for similarity to time traveler DNA.
Given enough time, you're going to end up with a very homogenous population. At least until some virus that the time traveler’s genes and nanos can't handle comes along. Then they probably all die due to a lack of genetic diversity.
The above is based on a lot of assumptions. Most of the problems can probably be resolved by changing a few items and adding in some handwaving.