I've made several canes. Initially, it was mainly as a turning exercise for me, and was stimulated by watching Richard Findlay do a demonstration at Totally Turning back in March. But a couple of months ago my wife broke her hip and a practical consideration entered the picture. Here's a thread on the subject -
https://www.penturners.org/threads/canes.154818/.
By the way, there are two kinds of canes - functional canes for those who need assistance in walking, and stylish canes that are mainly fashion accessories. Findlay produces 'swagger stick' canes mainly using oak stock. I think the main difference is the design of the handle - functional canes tend to have a handle that the user can actually grasp, while 'swagger sticks' tend to have decorative handles.
I've used ash, maple and oak. For the maple and oak versions, I laminated two pieces together to create a blank with a large enough cross section to be able to turn a shaft of the desired diameter. If the material is knot-free and with fairly straight grain, and if this is done with a good glue and the joint is clamped until the glue has fully cured, I don't think the fact that the blank is laminated will reduce the strength, and it might actually make the cane a bit stronger.
I use a lathe with a short (18") bed; however, when allowances are made for chucks and other fixings, the maximum length I can practically turn is more like 11-12 inches. As a result, the best approach for me is to make the cane shaft in three segments and then glue them together using tenons and drilled mortises. And if I want to include a coupler so that the cane can be broken down for storage or travel, it's best to plan on four segments. For strength, I make drill the mortises fairly deep - 1 1/2" - 2", and use tenons that very nearly fill the full depth of the mortise. Also, I make the mortise and tenons fairly large - around half the final diameter of the shaft.
Using commercial dowel stock for the shaft eliminates the lamination issue, and also the concerns about joining short segments to create the shaft. I presume that you are planning to also make the handle from commercial dowel stock, so the only issue will be how the handle joins the shaft. I opted to have a tenon on the end of the shaft that fits into a drilled mortise in the handle, and I've turned both from flat stock. I've also reinforced that joint on functional canes with a second dowel, made from either wood or fiberglass rod.
Would love to see a followup posting on your experience in cane making.