It's usually not worth your time to worry about natures when playing the main story, as long as your nature doesn't lower physical attack or speed. If you really want to choose between adamant and jolly, I think adamant is better. Weavile's base speed is already faster than almost everything, and it can outspeed those last few opponents by having a higher level or more EVs. Its STAB moves (namely night slash and ice punch) have relatively low base power and it doesn't like missing a 2HKO, so investing in physical attack is more likely to help.
Just in case you're wondering, a Weavile with 16 IVs, 84 EVs, and a neutral nature needs to be level 51 to outspeed Cyrus's Crobat, level 47 to outspeed Cyrus's Weavile, level 42 to outspeed Volkner's Jolteon, level 53 to outspeed Lucian's Alakazam, and level 51 to outspeed Cynthia's Garchomp.
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