My guess is that it won't matter at all if you do well in your masters. Satisfy the following three criteria (where 1 and 2 will lead to 3):
1. Take lots of pure math courses at a high level and do well (real, complex, functional, PDE, differential geometry, topology, etc...)
2. Do interesting research or advanced reading courses with profs
3. Get good recs.
All of these are possible to do after undergrad. I did my undergrad at a top ranked math/ applied math department, and I knew applied grad students who started doing math after undergrad.