... Adrian V, ruled only a very short time. The latter died at Viterbo on 18 August, 1276, having been elected on the preceding 11 July. In a consistory of cardinals, he had spoken of an alteration in the decrees of Lyons concerning the papal conclave, and had suspended them temporarily. After the death of Adrian V, the conclave in Viterbo was protracted, in consequence of which disturbances broke out in the town, thus hastening the election, so that in the week following 13 September Petrus Juliani, Cardinal-Bishop of Tusculum, was chosen pope, and crowned as John XXI (really XX) the following Sunday (20 September). The new pope wished forthwith to arrange the rules for the conclave. In the Bull "Licet felicis recordationis", ratifying his predecessor's decision, he also suspended with the consent of the cardinals the decrees issued at Lyons, and declared his intention of issuing in the near future the new regulations. http://www.newadvent.org/cathen/08429c.htm
Adrian was elected but likely never enthroned - that his name is traditionally catalogued in papal enumerations is of no particular consequence, as succession was achieved by election of a successor almost immediately after his death. Whether or not he was a layman is likewise a non-issue, as he'd not have been the first (and, if he was, likely not the last) person raised to the presbyterate and onward to the episcopate from the lay state upon being elected to a hierarchical office.
There is no patriarchal line, in any of the Churches, of which I'm aware that has not suffered breaks of time - sometimes prolonged - between the death of a sitting incumbent and the seating of a successor. Such delays have been occasioned by wars, schisms, murders, - you name it, they've happened. So, while far from being a staunch advocate of papal primacy in the form that it's taken, I suggest that speculating a loss of succession over Adrian is hardly a worthwhile pursuit for any purpose other than merely argumentation for its own sake. I'd expect more of some posters to this thread.
John XX/XXI, was elected within a month after Adrian's death and ratified the promulgation that Adrian had proposed relative to the conclave. Note also that the changes to be made were not substantive. They were intended to do away with the praxis of progressively restricting the sustenance afforded electors, to hasten elections and avoid some of the prolonged conclaves that had marked prior elections.
In 1271 the election that ended with the choice of Gregory X at Viterbo had lasted over two years and nine months when the local authorities, weary of the delay, shut up the cardinals within narrow limits and thus hastened the desired election (Raynald, Ann. Eccl., ad ad. 1271). The new pope endeavoured to obviate for the future such scandalous delay by the law of the conclave, which, almost in spite of the cardinals, he promulgated at the fifth session of the Second Council of Lyons in 1274 (Hefele, Hist. des Conciles, IX, 29). It is the first occasion on which we meet with the word conclave in connection with papal elections. (For its use in English literature see Murray's "Oxford Dictionary", s.v., and for its medieval use Du Cange, Glossar. med. et infimæ Latinitatis, s.v.) The provisions of his Constitution "Ubi Periculum" were stringent. When a pope died, the cardinals with him were to wait ten days for their absent brethren. Then, each with a single servant, lay or cleric, they were to assemble in the palace where the pope was at his death, or, if that were impossible, the nearest city not under interdict, in the bishop's house or some other suitable place. All were to assemble in one room (conclave), without partition or hanging, and live in common. This room and another retired chamber, to which they might go freely, were to be so closed in that no one could go in or out unobserved, nor anyone from without speak secretly with any cardinal. And if anyone from without had aught to say, it must be on the business of the election and with the knowledge of all the cardinals present. No cardinal might send out any message, whether verbal or written, under pain of excommunication. There was to be a window through which food could be admitted. If after three days the cardinals did not arrive at a decision, they were to receive for the next five days only one dish at their noon and evening meals. If these five days elapsed without an election, only bread, wine, and water should be their fare. During the election they might receive nothing from the papal treasury, nor introduce any other business unless some urgent necessity arose imperilling the Church or its possessions. If any cardinal neglected to enter, or left the enclosure for any reason other than sickness, the election was to go on without him. But his health restored, he might re-enter the conclave and take up the business where he found it. The rulers of the city where the conclave was held should see to it that all the papal prescriptions concerning enclosure of the cardinals were observed. Those who disregarded the laws of the conclave or tampered with its liberty, besides incurring other punishments, were ipso facto excommunicated. http://www.newadvent.org/cathen/04192a.htm
The stringency of these regulations at once aroused opposition; yet the first elections held in conclave proved that the principle was right. The first conclave lasted only a day and the next but seven days. Unfortunately there were three popes in the very year succeeding the death of Gregory X (1276). The second, Adrian V, did not live long enough to incorporate in an authoritative act his openly expressed opinion of the conclave. Pope John XX lived only long enough to suspend officially the "Ubi Periculum".