Advanced lipid testing such as NMRs etc aren't so easily available here in New Zealand, so I'm pretty much stuck with the basics, although a reasonable amount can even be implied by the basic stuff. For example, Dave Feldman gave an interesting talk at Low Carb Breckenridge 2018 about remnant cholesterol, which seems to correlate roughly with triglyceride levels. Then there's LDL-P, which has a striking inverse relationship with TG:HDL ratio (source).

As far as how long things take to change, I remember reading somewhere (possibly Grant Schofield's blog again but not 100% sure) that trigs can correct in a matter of weeks on LCHF, while HDL tends to improve more slowly over a period of months. I can't really offer anything on LDL-C beyond what you will have already picked up from Ken.