Nothing new, but returning to the mystery of the origins of Mme Jules Thibaud...
Some refs give this rose's date of introduction as 1881 and most declare it a sport of Mlle Cecile Brunner.
The date is Mlle Cecile Brunner's but whether Mme Jules Thibaud is a sport of that rose is not certain. Either way, this early date muddies the waters considerably. (See The Joy of Roses entry for example.)

To date, our first sighting of it in print is in 1949, where Mme Jules Thibaud is noted in a minature rock garden exhibit by an Essex nursery that specialised in minature plants.
By the mid 1950s Graham Stuart Thomas was listing it in his Sunningdale Nursery catalogues, where he made a point of saying he had been unable to trace its origins.
It was imported into Australia from Sunningdale by Deane Ross of Ross Roses and was also offered by NZ nurseries who may have also received it from Sunningdale, or from Ross Roses, or from another source.
It does not seem to be in the USA and I have not seen it mentioned growing in other parts of the world.
Plants we saw in one garden in England labelled Mme Jules Thibaud were indistinguishable from Mlle Cecile Brunner but this does not mean it is not grown there.

To my mind, Deane Ross's notes about this rose from 30 years ago are still the best summary of what we know of it.

"....Gathered around Cécile Brunner like an entourage is a cluster of variations and look-alikes often confusing to the newcomer to the world of heritage roses....I will turn now to a less known rose - ‘Mme. Jules Thibaud’ - which is consistently described as a sport of ‘Cecile Brunner’. For several years I went along with this statement, but all the time it did not seem to ring true. The blooms were different, being somewhat larger. More especially the open petals reflex into parallel tongues - in exactly the same way as ‘Perle d’Or’. It seems more likely that ‘Mme. Jules Thibaud’ is a sport, not of ‘Cecile Brunner’ but of ‘Perle d’Or’. Everything points that way. My earliest reference to ‘Mme. Jules’ is in the Sunningdale catalogue of about 1958, prepared by Graham Thomas where he says that it is “... a sort of ‘Cecile Brunner’. By 1962 when Graham Thomas wrote “Shrub Roses of Today”, he said “....a sport of Cecile Brunner”. Was this a typographical error or did he reconsider the origin of ‘Mme. Jules’. And of course, once a statement has been made by such an eminent authority as G. S. Thomas, subsequent authors copy it without question."
Deane Ross, Mlle. Cecile Brunner and Her Entourage.
The Australian Rose Annual 1994 p54. 
First printed in the Historic Rose Journal, March 1993 p13.

I will start a new comment as the older discussion was of considerable length, as is this comment.

Because of the various spellings of Thibaut / Thibault / Thibaud, I had a look through the Roseraie de l'Hay 1900 catalogue looking for similar names but only found:
p58. 'Mme Thiebaut Sine'. Leveque 1886 H.R.
p58. 'Mme Victor Wibaut'. E. Verdier 1854 H.R.
p59. 'Mlle. Clementine Ribault'. Ribault 1885 H.R.
p64. 'Pierre Liabaud'. Liabaud 1887 H.R.
p71. 'Ch. Reybaut' (inconnu) Tea.
p91. 'Mme. Andre Theuriet' l'Hay 1899 H.T.
....so no joy there.

Because of the lack of 'Mme Jules Thibaut /Thibault /Thibaud in the literature, I am wondering if if this could have been another of Nancy Lindsay's renamings. Looking at some dates:

1935 Allyson Hayward [in] Norah Lindsay. The Life And Art Of A Garden Designer
p190. [Norah visiting Johnny Johnston at Serre de la Madone in France] it has become twice as complicated as it was.... and too many pots in tiny gardens for my taste......
p195. At present there is nothing but red geraniums.....if he would only plant roses and jasmine.....

1946. Graham Stuart Thomas, Cuttings From My Garden Notebook
p146. James Russell and I [see 1959 note below] paid our first visit to Miss Lindsay's garden on June 30, 1946. The address was Manor Cottage, Sutton Courtney, Abingdon, Berkshire..... three of her main hunting grounds were Hidcote Manor (She was a friend of Lawrence Johnston) and ...... The thought-provoking names poured from her. For some years I was frustrated by these names because I could not find them in any of the old French books in my possession, nor in the Lindley Library. At length the reason dawned on me. Finding an unknown rose in an old garden without a name she let her fancy run free and coined a name for it. There were many others, all since resolved.

1946-47 RNRS Historic Rose Journal No. 37, 2009
Charles Quest-Ritson. ....there is a copy in the RHS Lindley Library of the slightly longer list that he [GST] issued the following year in 1946-47

1948 Allyson Hayward [in] Norah Lindsay. the Life And Art Of A Garden Designer
p243. Norah [mother of Nancy]] Lindsay died of cancer of the kidney on June 20, 1948.

1948. Graham Stuart Thomas - [in] The Graham Stuart Thomas Rose book
p6. The year 1948 was spent mostly in consolidating my own plants and getting to know the numerous varieties I had collected.

1949. Allyson Hayward. The Roses of Norah and Nancy Lindsay. [in] Rosa Mundi 2010 Vol 23, No. 2, p24
By approximately 1949, T. Hilling and others were carrying the [Nancy's] roses, and Nancy also began selling them directly from her Manor Cottage Nurseries. She produced a nursery catalogue titled Shrub Rose List describing her rose offerings

1949. The Gardeners Chronicle
p49 Long Barn Gardens, [Mulberry Green, Harlow, Essex] set out their attractive miniature rock gardens in pans, together with specimens of the Rose 'Madame Jules Thibaut'.

1958 Allyson Hayward [in] Norah Lindsay. the Life And Art Of A Garden Designer
p268 Lawrence Johnston 1871-1958. A close personal friend of Norah who often stayed with him at Hidcote and Serre de la Madone in France, he left the latter to Norah's daughter, Nancy.

1959
A November 20, 1959 invoice from Sunningdale Nursery shows the landscape manager of the nursery was James Russell and nursery manager was Graham Thomas. (See 1946, p146 above)

1962, January 20
An Australian nursery imports 'Mme Jules Thibaud' from Sunningdale Nurseries, U.K.

1963 Shrub Roses of Today
p147 'Deane Ross. Mme Jules Thibaud'. A sport from 'Cécile Brunner' very near to it, but of peach colour rather than pink. I have been unable to trace its origins.

.........................
Other references which may or may not be relevant:
2009 RNRS Historic Rose Journal No. 37
p9. Bunyard travelled widely in France and Germany looking for roses to add to his collection.
p10. Graham Stuart Thomas may have bought Bunyard's collection

Vita Sackville-West was a lifelong friend of the Lindsay family.

Somewhere I have read that Nancy Lindsay's father left her 100 pounds a year, and her uncle Peter (Norah's brother) bequeathed her Manor Cottage. She used the rear of the cottage as a small nursery. I didn't scribble down the name of the nursery but seem to remember it was something like the Cottage Nursery. She did put out a catalogue and if anybody can find one or two of these, they would be most interesting.
......................

It seems that we should call upon our English Heritage compatriots for any knowledge they may have of 'Mme. Jules Thibaud'.

Can any Gertrude Jekylll aficionado help me solve what Graham Thomas is talking about in this article. He is reviewing Gertrude Jekyll and Edward Mawley's 1902 book.

1974 The Rose Annual
p45. Graham Thomas. Miss Jekyll's Roses.
"at the end are lists of recommended roses; they contain many still grown today; ....Clair Jacquier' and 'Alister Stella Gray'; Hybrid Sweet Briar. 'Perle d'Or' is there but her photograph shows another rose."

I have the 1983 reprint of this book, but for the life of me, I cannot find the photograph of the 'Perle d'Or' substitute rose. If anyone has the original publication, I would love to see a scan of the photo.