Kate,
The earliest record I can find for Dineley in Northern Ireland, is a Death Record for a Josiah Dineley, suggesting a birth date of around 1823, but the nearest match to this that I have is a Josiah baptised in 1820 in Worcestershire. This could be right, but we would need more evidence to be sure. This differs from the date given on IGI reference (Abt 1816) to the Josiah Dineley/Mary Williams marriage in 1840 in Clogher, Ulster Province; but since this is not referenced I wouldn't rely on it.
I cannot find the Birth Record you mention for Andrew Thompson Dineley (although there are several later people of the same name), but I did find this,
Name Frederick William Dynley
Birth date 22 Oct 1873
Birth place Belfast, Antrim, Ireland
Father name Andrew Dynley
Mother name Sarah Shaw
There are also records for a Josiah Dineley and Ellen Jane Duncan having children around the same time. Interestingly, the couple seem to have had a child baptised in Scotland in 1873, so may have moved out of Ireland,
Name Josiah Dineley
Registration district Belfast
Record type DEATHS
Registration date -
quarter and year 1867
Estimated birth year 1823
Age 44
Father Josiah Dineley
Mother Ellen Jane Duncan
Eliza Dineley b. 15 Oct 1865, BELFAST,ANTRIM,IRELAND
Josiah Dineley b.20 Apr 1868, Ireland
Agnes Dineley b.05 Nov 1869, Armagh, Ireland
Herbert Dineley b.26 Jan 1867, BELFAST,ANTRIM,IRELAND
William Dineley b.16 Jul 1871, Ireland
Alfred Dineley b.24 APR 1873, Port Glasgow, Renfrew, Scotland
There is also a record for a Rachel Dineley, giving birth to a Jane McCready in 1866. Together this grouping MAY suggest that your Andrew had siblings called Josiah and Rachel.
Name Jane Macready
Christening place 390, BELFAST, ANT, IRE
Birth date 23 Apr 1866
Birth place Belfast, Ant, Ire
Father name Samuel Macready
Mother name Rachel Dineley
At first site, it seems plausible that the Dineleys that can be found in Northern Ireland from the 1850's onwards are from a single root, Josiah Dineley, and the date of birth around 1823 could allow for him to be descended from one of the grandchildren of Josiah Dineley of Peopleton, who died in 1790. He in turn was descended from the Dingley's of Charlton/Cropthorne. There is also the coincidence of the first name Josiah.
Finally, there was a John Dineley (grandson of Josiah) who married a Sarah Thompson in Warwickshire in 1820, but their children do not include a Josiah and the chronology of their other children do not suggest that one could be fitted in. Sarah died in 1830.
I certainly cannot disprove the potential link and the availble circumstantial evidence seems to support, rather than go against, it being likely. There remains the option that a family with a surname similar, but different, to Dineley (e.g. Denly, Dinlii, or Donelly) changed their surname (by accident or design) prior to 1850 and there is no link to Cropthorne at all.
Maybe if you have some more information to work with I could have another look, but I don't have much access to Irish Records. Do you have occupation details for your Irish family, or family anecdotes that might help? These may give a clue as to why the family moved across the sea.
I hope this of interest, even if it does not provide an answer.
Neil