How much would it cost, how long would it take, and what type of permit is needed to convert a bedroom into full bath and walk-in closet?
Much of this is dependent on your location. Cost, permit information, and other project related questions are going to be best answered by your local professionals. There is no one size fits most pricing schemes. The more competition there is in your area, the cheaper prices will be.
Cost is hard to say without a site visit but I can tell you this type of project can take anywhere from 4-6 weeks. You will need to have permits for the plumbing and electrical for sure, possibly framing and drywall but that depends on the city requirements on what they want to inspect during the job which varies from city to city.
1 month, building, pluming, hvac, and electrical permit.