With OED back I can add some info on Father Christmas. The first cite in OED is earlier than I thought (1658) but then there’s a long gap before the next cite (beginning of the 19th century) so my guess is that the earlier is a one-off and that the phrase didn’t take off until Victorian times.
b. Father Christmas n. the personification of Christmas as a benevolent old man with a flowing white beard, wearing a red sleeved gown and hood trimmed with white fur, and carrying a sack of Christmas presents. Father Time: see time n. 34b.
1658 J. King (title) Examination and Tryall of Old Father Christmas.
a1800 in Brand’s Observ. Pop. Antiq. (1813) I. 373 Lordings, in these realms of pleasure Father Christmas yearly dwells.
1860 Christmas Tree 189 ‘Tis now, when once more from his lair Old Father Christmas issues forth.
1860 Christmas Tree 190 Hail, Father Christmas! Come, and bring Thine ancient merriment and glee.
1864 R. Chambers Bk. of Days II. 740/2 Old Father Christmas, bearing, as emblematic devices, the holly bough, wassil~bowl, etc.
It’s quite a bit more complicated than that.
In 15th-century England a personification of Christmas already existed: a carol attributed to Richard Smart, Rector of Plymtree from 1435 to 1477, features a figure addressed as “Sir Christemas” and “Lord Christemas” who encourages everyone to drink and be merry at Christ’s birth.
Puritanism, on the rise in the early 17th century, discouraged Christmas merriment; this provoked Ben Jonson in 1616 to write Christmas his Masque, in which the same character introduces himself to the audience as ‘Old Christmas’, ‘Captain Christmas’ and ‘old Gregory Christmas’. Although he’s never specifically called ‘Father Christmas’ he is certainly a father, as his children Carol, Misrule, Gambol, Offering, Wassail, Mumming, New-Year’s-Gift, Post and Pair, Minced-Pie and Baby-Cake appear with him. Throughout the 17th century broadsheets and pamphlets appear featuring ‘Christmas’ lobbying for the old-style Christmas revelry; whether he is called ‘Father Christmas’, ‘Old Christmas’ or simply ‘Christmas’, he is manifestly the same person, as indeed is Dickens’s Ghost of Christmas Past. However, his function was not to bring presents, and he hadn’t any special interest in or for children.
But then, Christmas wasn’t ‘all about the children’ till well on in the 19th century. Indeed, medieval and Renaissance Christmas might well have been quite a scary and unpleasant time for children, combining as it did interminable church services (no fewer than three masses on Christmas Day) with one’s parents and other adults giving themselves up to Saturnalian revels typically including drunkenness, promiscuity, gambling and reversal-behaviour such as cross-dressing.
(At least medieval children in Britain, where St Nicholas had no special cult, were spared a visit from him on December 6th. That custom was very widespread and popular in Dutch- and German-speaking countries, though you have to wonder how popular it was with children as opposed to adults. Because although St Nicholas had a reputation for charitable giving, he also had a reputation for retributive physical violence - one widespread legend has him refuting Arius at the Council of Nicaea by knocking him cold, and a Dutch legend described how he convinced a Dutch abbot who was refusing to authorise the singing of prayers in his honour by appearing to him at night and thrashing him half dead. On 6th December he traditionally appeared to distribute gifts with a birch in his hand, and children who hadn’t been good or couldn’t say their prayers correctly would not only not get any presents but were put over the Saint’s knee and birched. Over the years he acquired companions to carry out this function for him; in northern France and Flanders ‘Père Fouettard’ (Father Flogger), elsewhere ‘Knecht Ruprecht’ (a Hagrid-style ogre), the Krampus (a goat-horned monster), or a black-faced devil (of whom the present-day Dutch Zwarte Piet is a much-softened descendant). These companions would variously threaten to flog naughty children, eat them, or put them in a sack and carry them off to hell. Only the very smuggest Little Goody-Two-Shoes child could possibly have looked forward to December 6th with unalloyed eagerness. It’s only in the last two centuries or so that the thrashing, threatening and coal-giving aspects were either dropped entirely or shrank to a joke-threat that even a child knew wasn’t really going to happen.)
In the mid-19th century, people in highly-industrialised, urbanised Britain felt the need for a cosy (but respectable - definitely no Saturnalia) family celebration in midwinter. With the new sentimentality about children, it seemed logical to revive the genuinely traditional Old/Captain/Father Christmas and give him the specific job of bringing them presents.
Incidentally, the British Father Christmas wears not a two-piece outfit and bobble hat like Santa Claus, but a long belted robe with a hood. It has been red since the late 19th century, but in earlier depictions (such as Dickens’s Ghost) it was more often green.