There's no single answer to how long it takes to learn Quran reading — it depends heavily on prior familiarity with Arabic letters, class frequency, and how consistently a student practices between sessions.
A complete beginner progressing through a structured Qaida with regular one-to-one classes will typically build core reading fluency over a period of months rather than weeks, with continued refinement of Tajweed happening well beyond that.
The biggest factor within a student's control is consistency. Two or three short, regular sessions a week — plus a little practice in between — tend to produce steadier, more lasting progress than infrequent long classes.
Rather than fixating on a fixed timeline, it helps to think in terms of milestones: confident letter recognition, then fluent reading with harakaat, then accurate application of Tajweed rules. A structured program moves a student through these stages in order.