I have no experience in the UKTA, but when I was in the ITF, the general timeline was gup testings every 3 months - so that was a minimum (which few people met) of 27 months from 10 gup (beginning white belt) to 1st gup (high red belt). Few people made the minimum for a couple of reasons - I knew few people who didn't miss at least one test sometime due to life events (job, family, school, etc.) that prevented them from being ready, and most classes didn't test at exactly three months due to holidays, availability of a testing instructor, etc.
So in theory (which rarely meets reality) the minimums are:
10th to 9th: 3 months
9th to 8th: 3 months
8th to 7th: 3 months
7th to 6th: 3 months
6th to 5th: 3 months
5th to 4th: 3 months
4th to 3rd: 3 months
3rd to 2nd: 3 months
2nd to 1st: 3 months
1st gup to I Dan: 6 months
Minimum training time: 33 months - so 2 years, 9 months - but then to add to the above interruptions in gup testing, the USTF in the Denver area (which is the ITF affiliate I started in) held group I Dan tests twice a year (so you tested with students from quite a few classes, instead of just in your own class), in (if I remember correctly) March and September - so if you weren't ready when that test was scheduled, you had to wait for the next one. I missed one color belt testing due to illness, and only had 4 months training when the first I Dan test after my 1st gup came around, and had to wait - I think it took me between 3 1/2 and 4 years to reach I Dan, and that was on the slow end of average.
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