Umbilicomancy

Divination by the umbilical cord; omphalomancy.

From New Latin vmbilicomantia (Agrippa), from Latin umbilicus the navel. 1995 C. Walker Encyc. Secret Knowledge 177: The arts of venamancy and umbilicomancy are divinatory systems linked with childbirth, which he [sc. Barthelemy Cocles] claims to have learned from his mother, a skilled midwife.

Not in OED. Original source lost!

Uranomancy

A rare word, recorded only once, in the 17th century, for divination by the stars; astrology; astromancy.

From ancient Greek ouranos heaven, the heavens, the sky.

The variant form included by Shipley is without other evidence. This form was quite possible in the 17th century, which had ouranography a description of the heavens.

1657 G. Starkey Natures Explication and Helmont's Vindication 16 (OED): All other natural pratical Arts, as Geometry, Astronomy, Uranomancy, Geography, Arithmetick, and the like.

1926 OED.

1955 Shipley Dict. Early Eng. (1963) 16: ouranomancy, uranomancy, the heavens. 1986 P. Hellweg Insomniac's Dict. x. 78: Uranomancy (or ouranomancy) The heavens.

Urinomancy

A rare term for the more common uromancy.
The 1979 citation is clearly referring to a 17th century text, however, the title was most likely originally in Latin, as was probably the whole book. It apparently was not one of Fludd's own texts. From urino- combining form of Latin urina urine.

1904 G.S. Hall Adolescence I. 116 (OED): The many centuries when urinomancy and urinoscopy vied with astrology.

1926 OED.

1979 Joscelyn Godwin Robert Fludd 64: [caption] This is part of the title-page of a treatise on 'Physiological Urinomancy, or diagnosis through the careful inspection of urine, in five books: in which the mystery, essence, species, causes and meanings of urine are discussed in a new way, different from the common teachings and scarcely known before.' Fludd devotes nearly two hundred pages to diagnosis through urine.

Uromancy
Medical diagnosis gained by inspection of the urine. First recorded in Agrippa where it is used derisively of physicians, along with drymimancy and scatomancy. This medical practise, roughly speaking, is still in use today, but it is not thought of as `divination' per se. However in earlier times, especially during the Middle Ages and Renaissance when physicians were informed by Galenian principles (i.e. the theory of the four humours), it was a very different diagnostic method.
As practiced in former times it was quite a simplistic. In a book by physician Thomas Brain entitled The Pisse-Prophet dating from 1655, we are told that, basically, diseases were categorised by uromancers into two types acute and chronic, and that dark coloured urine signified acute diseases, and weak, watery urine signified chronic diseases. Chemical analysis of urine was not performed, rather it was looked at concerning its "severall colours, parts, contents, substance, quantity, [and] smell". Brian points out that such urine colouring does not always signify acute illnesses. Here the physicians were using simple colour metaphor, red = hot, therefore = fever. However, Brian is probably simplifying things in order to discredit them. Certainly urine with any discharge signifies some malady, and clear urine is produced when healthy. This fact gave rise to the old proverb (as recorded in John Florio's Florios Second Frutes 1591) "I knowe no better phisick then to piss cleare, that so a man may bid a figg for the phisition".
Yet, uromancy was much more than merely diagnosing disease. Brian says (page 1):
  The vulgar sort are so strongly prepossest (by
  reason of their ignorance) the Physicians can
  discern (by the Urine) the Disease, the conception,
  the sex, the parties age, with many other such
  absurdities, that I fear it will be an hard matter
  to dispossesse them of that opinion.
Brain as quotes (page A2) a Latin proverb "often heard..spoken from the mouth of many a well-read and experienced man in Physicke, that (Urina est meretrix, vel mendax) the Vrine is an Harlot, or a Lyer..." The practice was also previously known as uroscopy, and, according to Bailey, as late as 1755, was "commonly called the casting of Waters". Rarely, termed urinomancy.

Etymology
From New Latin oromantia, uromantia, from uro- combining form representing Greek ouron urine.
Here -mancy refers to 'divination' in its weaker sense of 'sucessful conjecturing' rather than 'magical/supernatural prediction or insight'.

Variant Forms
Early modern English: oromancie, vromancie. Latinate forms: uromantica, uromantia. Modern: (influenced by the i in "urine") urimancy.

Citations
[1530 Agrippa De incertitudine & vanitate scientiarum atque artium: Hinc Scatomantia, Oromantia, Drimimantia, dicuntur divinationes seu prognostica medicorū ex stercoribus & vrinis deprompta.]
1569 Sanford tr. Agrippa Of the vanitye and uncertainties of atres and sciences lxxxiii, 145 b: Doo you not daylie see howe they [sc. physicians] goo about the Citie with ringed fingers, and dirtie gownes, with a sadde countenaunce, and alwayes pale, and with a quicke pace, for hope of veray vile gayne, renne from one Apothecaries shoppe to an other, seekinge and begging, if in any place a man wil offer them urine, or a ball of ordure to looke upon, & as the hooded vultures are beste nosed about deade bodies, so these mē, are best nosed of al about excremēts: which they say: Hippocrates was wont also to taste, that thereby he might the better preceive the notion of the sicknesse: whiche thinge also many doo attribute to Esculapius: who for this cause was of Aristophanes called Scatophagos, in whiche worde they be signified, which feede upon the superfluities of meates, which name afterwards was derived to all Phisitians, in such wise, that wée call them Scatophagians and Scatomanters, that is ordure eaters, and lookers on ordure. For this cause Scatomācie, Oromancie, Drymimancie, be called the diuinations or Prognostications of Phisitiās, gathered by ordures and urines.
[1603 Christopher Heydon A Defence of Ivdiciall Astrologie 9: Others also in derision name them, [sc. physicians] Scatomantes, Ouromantes, Drymimantes, to signify that their chiefest divination and living is by base matters; as by excrements, vrine, and the like, according to the verse: Stercus & vrina, Medicorum fercula prima.]
1625 James Hart The Anatomies of Urines I. v. 47 (OED): This parson being..reputed famous in vromancie.
1970 Man, Myth & Magic v. 658: Uromancy - by urine.

In Dictionaries
1708 Kersey Dictionarium Anglo-Britannicum: Uromantica, a Divining or guessing a [sic.] the nature of a Disease, by viewing the Patient's Urine.
1727 Bailey The Universal Etym. Eng. Dict. i: UROMANCY [ouromanteia, of ouron Urine, and manteia, Gr. Divination] the Guessing at the Nature of a Disease by the Urine.
1740 Dyche & Pardon New General Eng. Dict.: uromancy (s.) the divining, guessing, or forming a judgment by a perosn's urine.
1755 Bailey An Universal Etym. Eng. Dict. (16th ed.): UROMANCY [ouron, and manteia, Gr.] a divining or guessing at the Nature of Disease by the Urine.
1899 Century Dict. (1903) VIII: uromancy.. Diagnosis and prognosis of disease by inspection of the urine.
1899 New Sydenham Society's Lexicon V: Uromancy. Diagnosis or prognosis by obersvation of the urine.
1908 Funk & Wagnalls Standard Dict. ii.: uromancy..The art of determining the nature and result of disease by inspection of the urine alone; uroscopy. uromantia.
1912 Webs. New Int. Dict. uromancy [minor words list]
c1920 Cassell's New Eng. Dict.
1926 OED.
1955 Shipley Dict. Early Eng. (1963) 17: urimancy, urine.
1974 Mrs. Byrne's Dict. urimancy..fortunetelling with urine.
1986 Urdang (ed.) -Ologies & -Isms (3rd ed.) 212: uromancy Rare. a form of divination by studying urine. --uromantic, adj.
1993 McCormack Q&A 71: URIMANCY - urine.