No idea what an LCSW is, but yes, you do need a prescription. As for how to get it... Theoretically, anyone who can usually prescribe medications (which is, all medicine doctors, including general practitioners, and maybe nurse practitioners) is allowed to prescribe HRT (or directly refer to an endocrinologist, because whilst all doctors can prescribe hormones, endocrinologists knowledgeable in trans issues are the ones who can do it most effectively and safely and have the most experience regarding it) without further investigation, as long as you give informed consent and have been tested for physical contraindications.
However, many (most?) will first refer you to a professional, either a therapist (psychologist, sexologist or other) or a psychiatrist who will have the role of performing therapy and/or evaluating you: Are you truly transgendered? Are hormones really what you need? Do you have other mental health issues that need to be treated? Does your current state indicate that you might be likely to misuse (overdose) hormones or become instable? Do you have false expectations? etc.
Afterwards, you will be referred to an endocrinologist (or other doctor in a few cases I guess), possibly with a letter, and there, you should receive your prescription.
The whole approval process has a very varying length, depending on the beliefs of who is evaluating you, the unwritten laws that could exist (e.g. some have a set duration in their mind before which they will not refer, because they feel it's the time they need to be sure you don't have recurring depressions and your life is stable, and it's not just a phase, or sometimes just because they think transition has so many challenges that if you can't muster the wait, you can't go through it) as well as how fast your general practitioner refers you and how long the referral takes (administrative delays, waiting lists...)
It can be as short as not even one month for a few quick assessment and information meetings, but that is quite rare. It can also take up to two years if you have "co-morbid issues" to treat or the person evaluating you is really challenging, but that is quite rare, too. I think, from the moment you begin to see the person in charge of evaluating you, it's fair to expect an average of 6 to 8 months, maybe.
For me, it took a little over two years, but that wasn't just the approval process. It took me about 8 months, I think, to get my psycholgist to tell me I needed to tell my doctor, as well as to get my doctor to refer me to the psychiatrist (she wanted me to come out to my family first, which was hard for my shy self and took time). After about 6 months, I think, I first saw the busy psychiatrist. From that point onwards, it took exactly one year to get my referral for HRT.
The key to make it faster is to talk to your doctor and get the referral for evaluation fast. There are also, as I said, those "informed consent clinics" which won't really have you wait, but they aren't the majority, and there is also a possibility that you might get benefits from that therapy and evaluations.
The formulas vary. Some will prescribe just an antiandrogen to stop damage until evaluation is over, which is a great approach if you ask me.
