As was noted, success depends on the orchid, its health, growth stage, division location, root condition, and your growing environment and skills.
I reluctantly bought a well-grown 2-bulb ‘Chris’ Bronze Beard’ division in mid-July. By winter, it produced a small new leaf, followed by a full-size growth that flowered in the spring—an unusual result for me. Two key factors contributed: the new season's roots were emerging, and the seller's orchids were exceptionally well-grown.
Typically, even strong 2-bulb divisions of monofoliate cattleyas take three years to bloom. Three-bulb divisions may bloom the next year but could skip a year, while 4-bulb divisions often bloom the following year.
Bifoliate cattleyas are more temperamental; if not divided during active root growth, they can suffer serious setbacks or waste away. In my experience, smaller bifoliate divisions often need an extra year or two to bloom.
I would hesitate to buy a one-bulb division, especially if intentionally cut by the seller, as it raises concerns about their practices and motivations. At best I'd expect a two year to recover.