Well, here are some very personal observations about "The Bird and Baby, since I visited this pub in Oxford last summer. Please note that the entire sum of my great experience in British pubs amounts only to two weeks (by no means continuous), so I suspect that others of you across the pond could speak with greater experience.
It is, first of all, not a terribly fancy pub, in fact, not at all. It is plain but comfortable, with simple wooden backed benches and chairs. It has uneven floor boards and wainscotting painted brown, with plain yellow walls. I cannot recall cushions but it does have small heaters in each room. It is convivial without being ostentatious and would, I think, lend itself very well to simple meetings that occured regularly and often. That is, I suspect the Inklings could meet there and not make a horrendous dint in their academic paycheques. (I would not say the same of it now, as a mug with the name and insignia costs 20 pounds sterling, an outrageous sum to me, with which I refused to part.)
It is, also, just around the corner (essentially) from one of Tolkien's early houses in Oxford, so that ensured ease of access, which the Mitre, another of Tolkien's favourite pubs, did not.
I have to admit that I laughed much when I saw the sign with the eagle bearing the child. I could not help but think there must be some kind of joke there, with Tolkien writing of Gwahir carting Gandalf off. But then, I have a hunch that Tolkien had a stronger sense of humour than he is often given credit for.
The Lancastrian connection is interesting. If I find anything, I'll return.