As a specific example, J.R.R Tolkien delayed his enlistment until after finishing his university degree, and was commissioned as a Second Lieutenant afterwards. Was there an official process where university graduates immediately got qualified for officer commissions? Was there any relationship between Universities and the Military to facilitate this? Was this a process other countries replicated? Is this practice still in use?
If by "official process" you mean a universal and uniform process then the answer is "No." T.E. Lawrence (i.e. Lawrence of Arabia), who was a graduate of Jesus College, Oxford, received his commission as follows:
Shortly after starting at the General Staff, he was ushered into the august presence of General Henry Rawlinson. Rawlinson was about to leave London to take up command of British forces in Belgium, and Lawrence had been summoned to brief him on the newly updated Belgian field maps. Except, according to Lawrence, Rawlinson went apoplectic at the sight of his civilian dress, and bellowed, “I want to talk to an officer!” Since the Geographical Section now consisted of just two men, Lawrence was immediately bundled off to the Army and Navy Store, there to get himself fitted out as a second lieutenant, while the paperwork for his “commission” was hastily drawn up.
If you mean that having a degree was a requirement for a commission, also "no": Anthony Eden (later PM) was commissioned a 2nd Lieutenant at 18 and didn't receive his degree until after the war.
If you mean that individuals who did receive degrees would automatically receive commissions, then we can look at examples where this was not the case:
So from the roll of honour at Corpus Christi College, Oxford we have examples such as (emphasis added):
DIGBY, The Revd Lionel Kenelm (CCC 1902-1907)
Born Bramley, Surrey, 15 November 1883, 1st son of Reginald Digby, land-agent, and Caroline Grace [Boddington] of Geashill Castle, King’s County, Ireland.
Educated Harrow School 1897-1902.
3 Classics Moderations 1904; 3 Literae Humaniores 1906; BA 1906; 2 Theology 1907; MA 1909.
Leeds Clergy School 1907; Deacon 1908, Pretoria, Priest 1909, Birmingham; Curate of St Andrew’s, Bordesley, Birmingham 1908-1915; Rector of Tittleshall, Swaffham 1915-1918.
Military service WWI 1918 (service commenced 15 May 1918):
Private, 7th Norfolk Regiment. France. Killed in action at Faumont on 18 October 1918 (aged 34).
That being said, as you can see from this list, that was unusual. I count three privates, a corporal, and one sergeant (who then received his commission) on a list with about 100 names, all the rest of which are officers whether or not they finished their degrees. edit I had missed a large number of entries for "Service in the ranks before Commission", but the point stands that virtually all of them received commissions.
Edit The Lawrence quote is from Scott Anderson's Lawrence in Arabia.