Welcome to Rootschat, Diamond65. There seem to be quite a few bits of misleading information already on this thread (and using them isn't making the search easier). Just started looking but will start by posting what can be found in records.
Possible births for some of the children-
Birth: Frank Simpson, Jan./Mar.1893 Glanford Brigg dist., volume 7a page 679
Birth: Winifred Abigail Simpson, Jan./Mar.1897 Glanford Brigg district (Lincolnshire) volume 7a page 744
Birth: Doris Maud Simpson, July/Sept.1898 Glanford Brigg dist. volume 7a page 711
Birth: Arthur Simpson, Jan./Mar.1902 Glanford Brigg dist., volume 7a page 682
In 1901 this family are in Barton, Glanford Brigg registration district-
William Simpson 33 born Sheffield Yorks.
Mary Simpson 34 born Newcastle on Tyne
Willie Simpson 8 born Sunderland Co.Durham
Winnifred Simpson 4 born Barton on Humber Lincs
Doris Simpson 2 born Barton on Humber Lincs
Arthur Simpson 10 mo. born Barton on Humber Lincs
Any other records I can find show no connection to Ireland for William Simpson, wife Mary and children.
Census information Crown Copyright, from www.nationalarchives.gov.uk