Does Mangum get the Whig nomination for president or is he eschewed in favor of Clay? Would Mangum be more likely than Clay to win the election? A quick search on Wikipedia says the Whig convention was on May 1, two months after the accident and Mangum taking power.
Texas wouldn't be annexed by a President Mangum, and that might change things significantly. After all, the official casus belli for the Mexican-American War was that Mexican troops attacked a US patrol in the Nueces strip. Even if Polk is still elected and he tries to annex Texas early into his administration, it'd still probably delay the war a bit - and if Mangum (or Clay) is elected, then war should be avoided at least through 1848.