There are multiple definitions of custom.. some will work with you to change almost every thing about the instrument and some will offer a more limited set of choices.
My advice for what customizations make most sense
1. Wood
2. Left or right hand
3. Neck profile
4. Frets
5. Action
6. Pickup
7. Sound profile of the uke (the only luthier I know of who primarily focuses on sound profile and has a process is Mark Roberts, he has a few podcasts and articles out on the web if you want to research more but he offers choices for sustain quality, punchiness etc..) He is on my wish list to order from, I really do not know what to ask him.. I am still educating myself.
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many more options possible..
Speaking from personal experience here is what I know (please correct me if I make a mistake), by no means an exhaustive list and you can always reach out to the luthiers and discuss before ordering.
Folks like MyaMoe (multiple second hands in the past including the banjo) and Beansprout (on my list) will make a uke with limited number of customizations but an amazing price for the quality of the instrument.
Some one like Tinguitar (I have a soprano and super tenor) or Wunderkammer (
http://www.wunderkammerinstruments.co.uk, soprano) or Cripple Creek (soprano and concert) will do a lot more customization on the spec of the instrument. Their prices are very competitive for what is offered. Highly recommend (I have yet to receive my Wunderkammer but I have only read positive reviews..)
From here on the list is high end and even higher quality (in terms of aesthetics, the difference in sound quality is not as stark but there is a difference), some of the best ukes money can buy.
The higher end uke makers such as Beau Hannam will offer a lot of customization for a ukes that will make you drool.. heck he even built a fretless uke
Steve Grimes has standard models with some standard customization, though nothing is out of his reach if you can discuss it with him.
Hive (Jake Maclay) has standard customization on his site and a 2 year wait list (when I booked a year back)!
Pepe Romero also builds custom ukes but I think you have limited input on customizations beyond choice of wood and there is a 2 year wait list. He has a unique bracing style which gives his ukes a unique sound.
David Ingalls from Ono ukes make custom ukes and has been always pushing his design and build. Can't go wrong with him! Also, very competitive prices.
Tyde music makes some very interesting looking ukes (check out their instagram) and do offer custom inlays and aesthetics.
Manny Halican builds some amazing ukes but I am not sure if he customizes or he belongs in the list below.. I have yet to order from him but I have played his ukes.
Then there are luthiers who make hand made ukes but you have limited or no input and if you are lucky you can get one of their ukes..
Moore Bettah (Chuck Moore)
Les Stansell
Pegasus (Bob Gleason)
Ken Timms (only sopranos in Martin style but the best..)
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If you are open to second hand ukes, Compass Rose and Collings are great ukes as well and used to offer custom ukes..