This is a very difficult question to answer on documentary evidence. A quotation from Schrijver (2014) Language Contact and the Origins of the Germanic Languages, pp. 123:
[I]t is unclear when Dutch lost aspiration. The absence or presence of aspiration in Dutch p, t, k is not indicated in spelling, nor has it ever been. We know about non-aspiration only by listening to present-day spoken Dutch and by studying modern dialect descriptions that care to mention this feature, which do not go back in time beyond the twentieth century.
That gives us too wide a band - could aspiration have been lost in contact with late Latin in the first few centuries CE, or with Old French around the 10th century CE, or with Early Modern French in the 17-18th centuries? Schrijver also then goes onto mention the band of Westphalian German dialects which also lack an aspiration contrast in their fortis-lenis pairs.
However, using dialectal evidence of i-umlaut vs "spontaneous" fronting across the some but not all of the many modern dialects of Dutch and Frisian (spontaneous fronting being a feature of "Western" Dutch, on which modern Standard Dutch is based), most notably the fronting of long u: to /y/, as well as similar vowel patterns across Picardian, East Walloon and West Walloon (all Langues d'Oïl), Schrijver comes to the conclusion that:
Spontaneous fronting accordingly is an Old French substratum feature in Dutch.
As Picardian in the Middle Ages is fairly well attested, it gives a later bound of the 11th century, and the Romance-to-Germanic shift of the population adopting is put at 8-9th centuries CE.
Additionally, the lack of the High German Consonant Shift (HGCS) also put in a later bound, sometime around the 9th century (from when the earliest identifiable manuscripts in High German are dated).
The rest of the tale concerns dialect mixing and levelling of the fronted vowels, which is incidental to [and happens presumably after] the loss of aspiration as these Romance-speakers adopt a Germanic form of speech.