The evolution of "thereupon" is examined in detail in a study of historical corpora by Aune Osterman titled "There compounds in the history of English" published in Grammaticalization at Work, Studies of Long-term Developments in English.
Along with therefore, thereupon is the only other compound adverb [of the word "there"] that emerges in [Middle English] but continues into [Early Modern English] and up to the present day. The first four instances that appear in ME1[1150-1250] in the present corpus are all local in meaning: 'upon that or it'...
In ME3[1350-1420] local meanings are still foremost (58% [based on the Helsinki corpus when accessed]).
Osterman elaborates that in Middle English subperiod 3 (1350-1420) the word was sometimes used in medicinal recipes, as in this example translated to modern English:
Put paste or clay thereupon all over as told before, put burning hot honey evenly thereupon.
The first temporal meaning 'after that' is also cited in ME3, and according to Osterman's statistical analysis of the Helsinki corpus, temporal meanings were "by far" the most common senses of the word in Late Middle English through Early Modern English.
- 1420-1500 (55% of uses examined were temporal)
- 1500-1570 (61% of uses examined were temporal)
- 1570-1640 (80% of uses examined were temporal)
To get back to the question, these subperiods provided above give a good impression of how the meaning of the word shifted from being primarily locative to primarily temporal.
Note also that the Middle English 3 sub-period where the temporal sense is first found ranges from 1350-1420, and specifically 1400 if we believe the attestation provided by the OED, whereas in the Middle English 4 period, 1420-1500, the temporal sense was already "the most common." This means that it didn't take long after the temporal sense of "thereupon" entered use for it to become the dominant use, beginning the eclipse of the locative meaning that is observed in the question.
Moving forward, Osterman finds a new sense of thereupon in the late subperiod of Early Modern English (1640-1710), meaning "on that subject or matter, with reference to that/it/them, thereanent." These uses were common in law texts, as in this example:
Sixty Pounds over and above the Dutyes already charged thereupon without any Deduction
The conclusion of Osterman's analysis forms a solid summary of the evolution of "thereupon" from a locative term to a temporal one.
From the first local meanings thereupon develops meanings expressing temporal relations and after that abstract or figurative usages appear. Temporal meanings prevail from [1350 to 1640] with local meanings lingering alongside of them to the end of that subperiod. In [the last Early Modern English subperiod, 1640-1710], temporal and abstract usages are present.