Two things commonly hold up transfers to treatment facilities: overcrowding and detainers. If the treatment facility is at capacity, incoming transfers get backed up in county until a bed opens. That can take weeks or longer, depending on how the facility manages its population. There is no reliable way to rush it from the outside.
The other possibility is a detainer, which is a legal hold placed by another jurisdiction that has pending charges or a warrant against your son. Detainers can freeze a transfer entirely because the holding county cannot release him to another facility until the detainer is resolved. If there are any other counties with open cases or warrants, that may be what is blocking the move. You can ask the county jail directly whether there are any detainers or holds on his file.
On the time credit question: yes, every day he is sitting in the county counts. Time served in county jail is credited toward whatever sentence was imposed, day for day. He is not losing time while he waits. That credit will be applied when the case is finalized and his actual sentence calculation is done.
Stay in contact with his case manager or attorney and ask directly whether a detainer is involved. That is the most useful piece of information to have right now.