No, there’s no mention of Muggle werewolves gaining magic.

Though it is possible for Muggles to become werewolves, there’s no mention of them gaining magical abilities when they do, nor is it mentioned that they become more sensitive to magic.

“Once a month, at the full moon, the otherwise sane and normal wizard or Muggle afflicted transforms into a murderous beast.”
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Most Muggles who are bitten by werewolves end up dying anyway, so there are comparatively few werewolves that are Muggles - most people who survive a werewolf bite are wizards.

In the late nineteenth century the great English authority on werewolves, Professor Marlowe Forfang, undertook the first comprehensive study of their habits. He found that nearly all those he managed to study and question had been wizards before being bitten. He also learned from the werewolves that Muggles ‘taste’ different to wizards and that they are much more likely to die of their wounds, whereas witches and wizards survive to become werewolves.
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There aren’t many Muggle werewolves, but there are some, so if they did gain magical abilities, it’s likely this would have been mentioned or recorded with the other information on werewolves.

Muggles usually die.

Here's a Pottermore quote I found on it:

He also learned from the werewolves that Muggles ‘taste’ different to wizards and that they are much more likely to die of their wounds, whereas witches and wizards survive to become werewolves.
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There are no specific mentions of a muggle ever becoming a werewolf, and there's also no mentions of a muggle exhibiting magical powers due to being bitten though it does seem that it is possible.