What is non-borrowed native English word meaning "calculate", "compute", "count"? What was the word for these things in Old English?
Reckon comes from the Old English recenian, meaning “to pay, arrange, dispose, reckon”.
The Old English roots for "number" were both "rim" and "tæl"/"talu" (as can be seen by entering "number" and translating to Old English here).
Using that as a starting point,
tally: (v) calculate the total number of;
reckon: (v) establish by counting or calculation; calculate.
To see that the etymology of tally goes back to the Old English, consider words zala/zahl.
Zahl in modern German means "a number".
Zala meant "number" in the Old High German.
The fact that t/s sounds can be viewed as interchangeable and one can become permanently transitioned into the other can be seen from the Hebrew Tav/Sav. Modern Hebrew dispensed with Sav altogether and uses Tav both where Tav and Sav used to be.
Officially, "tally" is considered to have entered English in the mid-English by the way of "cutting"->"entering notches"->"counting". But the fact that its sound is so similar to "talu" seems too coincidental.
"reckon" came in as a variation on "rim".