Original poster: I agree with what many have posted. You almost certainly, but not perfectly certainly, have CTS.
The splint and cutting back on uke time is a start--wear the splint every single night for a month...wherever you stand on the resolution at that time is about how good it'll get. You might, unless there is a contraindication to doing so (e.g. you're on warfarin, you're already on an NSAIDs, have kidney disease, you have bleeding ulcers in the past, and on and on--a live doctor needs to judge this for you) might try Aleve or generic at 2 tablets (that's double the dose on the bottle, that's why you must be sure there's no contraindication) twice a day for two weeks scheduled, not as needed. You can judge it then; if still noticeable, see your doctor. Agree with poster below, if your doctor is unconcerned, you need another doctor. They'll order an EMG test, which is not painful (I do them for my patients, and on myself as I teach med students). It'll determine if you, in fact, have CTS or some other condition (such as the pinch nerve in the neck, as someone suggested). It'll also determine the degree of CTS that you have.
Untreated CTS, even if it goes away, will almost always come back, and eventually you'll have the 20 minute surgery to get rid of it. There's an 8% chance of recurrence, which is low. Predisposing factors for CTS are not only repetition (like uking), but are related strongly to heredity (mom, dad, bro, sister with it) and also if you have associated conditions like diabetes and thyroid disease. Sometimes the CTS reveals itself before the underlying disease, and hence a caring primary doctor is key if it persists past a month or returns to assure that you don't have a predisposing, treatable and important disease. This is why treating CTS with a chiropractor is not advised, as it's unproven in any clinical study, provides only temporary relief, and as chiropractors are not licensed to order lab tests, EMG tests, and prescribe medications, they cannot determine for you if you have the important underlying conditions.
Complications from CTS surgery from an orthopedic surgeon with experience (i.e. in practice for five years or more) are exceedingly low, and the horror stories about CTS surgery are by and large related to people who let the condition go untreated, damaging the nerve, and then having the surgery to prevent progression after the nerve is permanently damaged and then complaining about the surgery. CTR (R=Release) is not surgery "on the nerve", but rather around it to free the nerve. It's one of the safest surgeries available for any condition, and is hugely effective (when the person first has an EMG to know the condition exists; some unscrupulous surgeons do the surgery without the EMG, and when the patient continues to have symptoms only later to find out they didn't have CTS at all;, this is another, less common, reason for a disgruntled patient to report a poor surgical outcome). The recovery is quick, with office workers typically back to work in 1 week, and hand laborers back in 2-3 weeks. Laser surgery, touted by some, is unproven to open and release enough for long term efficacy, and the data is not in for recurrence rates on so called laser surgery. Traditional open, 1/2 inch incision surgery is so easy, that laser holds little advantage, and the potential disadvantage to having to have the traditional surgery in a few years. Further info on that would be through your particular surgeon if it gets to that point clinically.
Mahalo and good luck to you, Bob from CK1.