Expected to have been born in Birkenhead, and after Jim.

Cheshire Birth indexes for the years: 1921 to 1923
NUGENT John Richard Birkenhead Wirral MATHER BIR/294/96

NUGENT, JOHN  RICHARD      MATHER 
GRO Reference: 1921  J Quarter in BIRKENHEAD 
Volume 08A  Page 839

Death record and 1939 register show John R Nugent born 28 Feb 1921

The local paper reported:
NUGENT - JOHN RICHARD (JACK)
Josie and family would like to thank relatives and friends for their cards, flowers and kind messages of sympathy following their sad loss.  Sincere thanks to Doctor Partridge, Upton Group Practice for his caring support.  God Bless you all.

Newspaper cutting shows that Josie was still alive when Jack died in 2000, so this 2001 death cannot be correct:
Name: John Nugent
Birth Date: 19 May 1913
Death Registration Month/Year: Feb 2001
Age at death (estimated): 87 
Registration district: Birkenhead 
Inferred County: Cheshire 
Register number: C8C 
Entry Number: 85 

Jack must have died before 2000, and this is a better fit.
Name: John Richard Nugent
Birth Date: 28 Feb 1921
Date of Registration: Dec 1999
Age at Death: 78
Registration district: Liverpool
Inferred County: Lancashire
Register Number: C23D
District and Subdistrict: 0251C
Entry number: 173
This fits with a Birkenhead birth (instead of Oldham for the other).

Name: Alice E Green
Mother's Maiden Surname: Perrin
Date of Registration: Apr May Jun 1924
Registration district: W Derby, Lancashire
Volume Number: 8b
Page Number: 1177

An ancestry member (possibly John Nugent) records Alice's birth as 18 Feb 1924, and death 6 July 1966, which is consistant with other records.

Died before Jack married Josie.
About 1965 to 1970

Name: Alice E Nugent
Death Registration Month/Year: 1966
Age at death (estimated): 42
Registration district: Birkenhead
Inferred County: Cheshire
Volume: 10a
Page: 28

An ancestry member (possibly John Nugent) records Alice's birth as 18 Feb 1924, and death 6 July 1966, which is consistant with other records.

Probate:
NUGENT Alice Elizabeth of 25 Bassenthwaite Avenue, Noctorum, Birkenhead died 6 July 1966. Administration Liverpool 21 September to John Richard Nugent staff Electrician. £4476.