The OED, for the verb :
to make patient' or 'to be patient'
gives references to both the French pacient and the Latin patientem which are present in English at about the same time, but originally with different spellings.
1551 R. Robinson tr. T. More Vtopia sig. Eiii Patient iourself, good maister Freare (quod he)..and be not angry.
1561 T. Norton & T. Sackville Gorboduc iv. ii. F iij b Pacient your grace, perhappes he liueth yet.
Later, the spellings converge.
For the adjective :
Enduring pain, affliction, inconvenience, etc., calmly, without discontent or complaint; characterized by or showing such endurance.
the OED gives
c1350 Apocalypse St. John: A Version (Harl. 874) (1961) 4 (MED) His entent is to amonesten to be pacient, for þorouȝ þouȝt alle þe tribulaciouns..ben hard & stronge forto suffren.
The OED gives several references in the French spelling then it appears to merge with the Latin spelling with Shakespeare :
1600 Shakespeare Merchant of Venice i. iii. 108 Manie a time..you haue rated me..Still haue I borne it with a patient shrug.
The OED also gives a somewhat different shade of meaning to the adjective :
Able to wait calmly; quietly expectant; not hasty or impetuous.
and references this back to the Wycliffe bible of 1382, again with the French spelling :
1382 Bible (Wycliffite, E.V.) (Bodl. 959) (1965) Ecclus. vii. 9 Betere is a pacient man þan þe enhauncynge hymselue
They list yet another meaning
Undergoing the action of another; passive; (also) achieved or acquired inwardly. Chiefly in contrast with active, agent. Obs.
and reference this to, first the French and then the Latin spelling :
a1425 tr. Guy de Chauliac Grande Chirurgie (N.Y. Acad. Med.) f. 76v (MED) Cause particler, pacient, or suffryng [L. Causa particularis patiens], was disposicioun of þe body as cachochimia, i. yuel chimez, & debilitacioun & opilacioun.
a1522 G. Douglas tr. Virgil Æneid (1957) II. 22 The ayr and the fyr is actyve, and the watyr and the erth patient.
It is interesting to see the scope of the concept - first patience, then the acceptance of ministering from another, then the idea of a 'patient' emerging.
Then, the noun :
A person receiving or (in later use) registered to receive medical treatment, esp. at a particular establishment or from a particular practitioner; a person staying in a hospital for medical treatment.
is also referenced with both spellings (French and Latin) :
1387-95 Chaucer Canterbury Tales Prol. 415 He kepte his pacient a ful greet deel In houres by his magik natureel.
a1500 tr. Lanfranc Sci. Cirurgie (Wellcome) f.28 (MED) If þe mater be mechill, the mater shall be þrest lytill and lytill, namely if þe pacient be febill.
1547 A. Borde Breuiary of Helthe i. Proheme f. iiii Chierurgions ought..nat to be boystiouse aboute his pacientes but louyngly to comfort them.
1623 Shakespeare & J. Fletcher Henry VIII iii. ii. 41 He brings his Physicke After his Patients death
firstly with the French and then (again in Shakespeare) the Latin spellings.
It is interesting to see both the origins (French and Latin) come into English simultaneously, with, initially, their respective spellings. And also to see that slightly different meanings are being brought in as well.
I think it would be a rather large study to sort out the several subtleties of meaning, perhaps the several localised or dialectic shades of usage, and to track the merging of the two spellings.
But it is an interesting example of how English formed.
All references from OED