Height and Weight — How to write them when abbreviations are not used

Your suggested punctuation system for compound or mixed measurements seems perfectly reasonable—and it finds support in at least one style guide. From Words Into Type (1974):

Dimensions, weights, and measures. Commas are unnecessary in the punctuation of phrases denoting dimensions, weights, and measures.

five feet seven inches

4 lb 3 oz

5 hr 10 min

His age is 6 years 4 months 12 days.

On the other hand, the house style at the publisher I work for has long required a comma between hours and minutes when reporting timed results of performance tests—such as "4 hours, 39 minutes." The Chicago Manual of Style, 15th edition (2003) doesn't directly address the question of how to handle such mixed measurements, but one entry happens to show how the University of Chicago Press deals with them:

9.45 Seconds and dates included. A variation of the twenty-four-hour system [of expressing time] shows hours, minutes, and seconds separated by colons; it also shows fraction of a second following a period. This format may be preceded by an ISO-style date [cross reference omitted].

09:27:08.6 = 27 minutes, 8.6 seconds after 9:00 a.m.

1999-05-10-16:09:41.3 = May 5, 1999, at 9 minutes, 41.3 seconds after 4:00 p.m.

In both of these instances, Chicago is centrally concerned with translating the "twenty-four-hour system" variant examples (first without and then with a preceding ISO-style date) from numerals into words; but in doing so, it consistently separates the minute term from the second term with a comma, contrary to the Words Into Type recommendation.

Either the Words Into Type approach or the Chicago approach, if consistently used, should achieve the primary goal of any such system of punctuation, which is to convey the relevant measurements clearly while posing a negligible distraction to readers.