Only Poole I can find was Sarah Elizabeth Poole Mar.qtr.1844 Scarborough who in 1861 census is in Southwark with mother Ann. The Henry Poole with James Logie in 1881 is, I think, Sarah's brother. He's said to have been bc.1835 Manchester and in 1841 at Manchester is a Henry Poole 6, son of Thomas and Ann. (In 1851 he is in Ross, Herefordshire.).
So 'Lizzie' in 1871 could be Sarah Elizabeth Poole but cannot find a marriage to James Logie nor can I find a birth for Jasper under Logie or Poole but there are entries for Leon Lionel and Ada Adine mmn Poole at St Saviour.
Annette