Oh thanks everyone for being so busy in my absence!
Regarding the witness to Robert Jnr and Alice's wedding, WILLIAM BURRELL:
It has been suggested that Alice might have been married previously to WILLIAM LAWS - the entries (cited previously) are thus:
24 Aug 1774 William Laws married Alice Burrel, St Hilda's Church, Hartlepool
Witnesses: Robert Shadforth, Jane Hastings
12 Feb 1776 Burial: William Laws, taylor (sic), St Hilda's Church, Hartlepool
Now, I have estimated that at this time Robert Shadforth Junior would have been about 16 yrs old, so presumably the witness to this wedding was Robert Shadforth Senior.
Now, if this is our Alice, ALICE BURRELL,, there are records of two baptisms at appropriate times for her and possibly her brother, both at St Hilda's:
19 Sep 1748 Alice Burrell daughter of Robert Burrell
10 Jun 1753 William Burrell son of Robert Burrell
So taking Trees' suggestion to look at the witnesses to the marriage of Robert Jnr and Alice, it could well be that WILLIAM BURRELL was Alice's brother. So far I can't find Ann Coulson though.