Nun born approx 1855 entered Gateshead Convent 1881+ Katherine McELROY.

[River Tyne Lass]

https://www.findagrave.com/memorial/250468742/catherine-mcelroy?createdMemorial=Yes

Doreen, I have now entered this Catherine on Find A Grave and have also put in a photo request for a memorial photo, if one exists.
I am a regular contributor to Find A Grave myself and I know that in my local cemeteries Nuns do seem to be listed on Convent memorials.
It's worth a try with this I think.
I will let you know if a response might be forthcoming if and when it comes.

[River Tyne Lass]

Doreen, I am sorry - I can leave the memorial on Find A Grave but I will have to remove the photo request.
I have just noticed it is recorded on the info about the cemetery that some sections will not have headstones and unfortunatey 'A' seems to be one of these.
I wonder if perhaps this Catherine was a resident rather than a Nun? 

[heywood]

... there is a death for a Catherine McElroy in 1936 Vol 8b Page 129 age 82 in Liverpool.  This would give birth year around 1854.

The burial entry that matches that death registration:-

Liverpool, Ford Cemetery, Catholic Burials, Public Sector S.V.
7 Jan 1936
Catherine McELROY
Age 82
Residence - St Augustines Home of 39 Yew Tree Rd.
Grave 1 A 8

These seem to be two separate addresses.
I think she may have lived at 39 Yew Tree Road but died in St Augustine’s Home.

[heywood]

There is a Catherine McElroy b 1852, Fermanagh in 1921 (perhaps a servant to a family named Lancaster).

1911 has a Katherine McElroy b 1857, Fermanagh who is a servant in Liverpool.

[Doreen Peacock]

Hello all you Wonderful People, here is a Summary of all the Red Herrings we had in the history of my Grandfather George Wm. BEST, McELROY SHIEL.
HIS MOTHER, WAS A SINGLE LADY WHEN SHE GAVE BIRTH.
SHE DID NOT HAVE ANY CHILDREN PRIOR TO GEORGE IN 1881
SHE LODGED WITH THE SHIEL FAMILY, WHO HAD A DAUGHER EMMA A. AGED 12  IN 1881.
CATHERINE DID NOT ENTER A CONVENT, NEITHER DID EMMA.
FATHER OF GEORGE WAS WM. HENRY BEST A MARRIED MAN WITH CHILDREN.
AFTER INFORMAL ADOPTION CATHERINE (KATE) GOT AN ASSISTED PASSAGE FROM PLYMOUTH TO COOKTOWN, QLD.
SHE MET HER FUTURE HUSBAND AND HAD THE FIRST OF FIVE CHILDREN WITH HIM IN 1883. (THIS CHILD WAS CALLED VERA INEZ LUCY MUSGRAVE, AND SADLY DIED1885.
THE NAME VERA CAME FROM A MIDDLE NAME OF MRS. SHIEL...THE NAME INEZ WAS PROBABLY THE NAME OF A CARER/HOUSEKEEPER THAT HELPED BRING UP CATHERINE AND SIBLINGS, AFTER THEIR OWN MOTHER DIED WHEN CATHERINE WAS 10 AND HAD TWO YOUNGER SIBLINGS, AND ONE OLDER, AT THE TIME HER OWN MOTHER DIED. PETER, HER FATHER WOULD HAVE TO WORK TO KEEP THEM ALL TOGETHER).

SECOND CHILD BORN TO NEW PARTNER,THE SAME YEAR VERA DIED.
HER LAST CHILD WAS BORN IN 1889.
SHE MARRIED THE FATHER OF HER AUSTRALIAN CHILDREN IN 1889, AND SADLY DIED IN 1891
SHE IS BURIED IN BUNDERBERG CEMETARY.
 
WM. H. BEST THE ERRANT FATHER OF GEO BORN IN 1881 FOLLOWED HIS WIFE AND CHILDREN TO QSL APPROX 1883. LATER SOME OF THE FAMILY RETURNED TO ENGLAND, AND ONE LIVED IN THE SAME TOWN AS ME WHEN GROWING UP. I COULD HAVE MET AND MARRIED A CLOSE BLOOD RELATIVE!!!!!
ANOTHER MEMBER OF THE FAMILY RETURNED TO THE TYNE AND WEAR DISTRICT, 40 MILES FROM MY HOME.   MANY, MANY THANKS FOR ALL YOUR INPUT, IT DID HELP CLEAR THE AIR, DROSS AND FALSE LEADS, AND WAS AN INTERESTING JOURNEY TRACING THE MYTHS AND SPECULATION THAT HAD BEEN FLYING AROUND THE FAMILY FOR GENERATIONS. THE TRUTH WAS HIDDEN IN THE PROVERBAL HAY STACK, AND GAVE US ALL A RUN FOR OUR MONEY, BUT WELL WORTH CLEARING THE POKIE PIES AND FALSE LEADS. IT MADE THE LITTLE GREY CELLS WORK A BIT HARDER; BUT YOU ALL GOT US TO THE FINISHING LINE ON THIS TRACING.....NOW I NEED TO CLARIFY WHO MY FATHER IS....   THANKS A MILLION.

[River Tyne Lass]

Thanks for posting this update Doreen, it is very kind of you.  It is always good to know how things turn out after debunking myths and speculation.  This has been a very interesting thread.
I am very happy for you in your success in finding things out.
Do come back if you feel we might be able to help you again in future.

[Doreen Peacock]

All of your help and advice was invaluable and very much appreciated to help focus on routes to continue the searches. Even the wrong routes were interesting and valuable lessons learned in researching for future adventures into the past. NOTHING IS EVER A WASTE OF TIME. THANK YOU ALL SO MUCH FOR THE TIME YOU ALL  DONATED TO ME AND MY FAMILY, IN  SOLVING THE MISSING LINKS IN MY HISTORICAL FAMILY TREE ON MY MATERNAL, GRANDFATHER'S, MOTHER'S AND FATHERS LINES.