You might like to check my
harmonica review pages for some ideas and suggestions.
There are two types of harp: diatonic and chromatic. Chromatics are almost always in C, and can play any key, but they're much more difficult to learn than diatonics. Chromatics have a slide button on the side.
Diatonics are the standard harmonicas you see in music store cases. Get one in C for sure. That means you can play in C on the uke, and play blues in G. Other popular keys are A, F and D (which give you blues in E, C and A respectively). G is also good - blues in D.
You probably won't need any flats or sharps unless you play blues in F - then get a Bb.
Forget the minor keys, Dorian tunings, etc. for now. But minor harps are also fun to learn and a bit funky.
If you think you want to pursue the harmonica, don't waste your money on a cheap throwaway. They're awful, seldom in tune and the reeds wear out too quickly. Buy something better - a Hohner Big River is a good buy and not too expensive. Or look at Lee Oskar, Hohner Golden Melody, Suzuki Bluesmaster, Seydel and so on.