Subword tokenization critically affects Natural Language Processing (NLP) performance, yet its behavior in morphologically rich and low-resource language families remains under-explored. This study systematically compares three subword paradigms -- Byte Pair Encoding (BPE), Overlap BPE (OBPE), and Unigram Language Model -- across six Uralic languages with varying resource availability and typological diversity. Using part-of-speech (POS) tagging as a controlled downstream task, we show that OBPE consistently achieves stronger morphological alignment and higher tagging accuracy than conventional methods, particularly within the Latin-script group. These gains arise from reduced fragmentation in open-class categories and a better balance across the frequency spectrum. Transfer efficacy further depends on the downstream tagging architecture, interacting with both training volume and genealogical proximity. Taken together, these findings highlight that morphology-sensitive tokenization is not merely a preprocessing choice but a decisive factor in enabling effective cross-lingual transfer for agglutinative, low-resource languages.