Having dealt with litigation in both varieties of jursidiction, I much prefer code over common law (the latter leads to the rent-seeking grift of providing search engines for sufficiently handy precedents, and a long phase of arguing over what the law may be, instead of starting from a shared understanding of the law and going directly to arguing about what the facts may be).
As to the actual question: for immediate relatives of english, perhaps icelandic (is)? https://en.wikipedia.org/wiki/Icelandic_language#:~:text=Tho....
In the indo-european context, lithuanian (lt) is said to be relatively conservative: https://en.wikipedia.org/wiki/Lithuanian_language#Old_Lithua...
If we don't require widespread oral use, https://en.wikipedia.org/wiki/Classical_Chinese (lzh) goes back a ways, but by current I think you do mean to require widespread spoken use. this also disqualifies (he) for the interregnum during which it was merely liturgical.
A basic issue seems to be that within a language family, different branches tend to conserve and innovate in different ways, making it difficult to pick "most conservative" out of any current members.
Did you have any languages in mind?