Your part of the Bayman family did indeed come from Kent but further back then Francis Llewellyn.William Bayman Married Margaret Francis on the 21.2.1831 redg. Beckinham, Kent.They had a son James born 12.1.1832 redg. Farnborough, Kent.He married Ann Swan 24.7.1853 in the Parish ch. St. Lukes, London. James and Ann had Ann - C.20.5.1854 M. John Clarke 26.12.1887 James - B 8.3.1856 M. Ida Sargent 25.12.1881 Margaret Frances - C.27.7.1857 M. Robert Johnson 11.7.1880 Eliza - B 25.3.1859 (my g-grandmother) M. Samuel Martin 25.9.1881 William - B 1860 Sarah Louise - B 1861 D 1862 Lavinia - B 1862 D 1863 Henry Swan - B 1864 M. Emma Halsey 5.8.1883 George - B 1866 M. Rose Halsey 26.12.1891 Francis John - B. 1868 M. Ann Rachel Walker 5.7.1891 Llewellyn William - B 1869 M redg Dec qtr 1893 Samuel Edward - B 1871 Harriet Eleanor - B 1873 Edward Percy - B 1874 The last three died aged 1,2and 0 respectively. All the above children were redg at Edmonton, Mddx.
You can tell me if Francis Llewellyn was the son of Francis John or Llewellyn William. I haven't found out yet.
Francis L. also had a son Ronald F. who had a son Richard who lives in London but maybe you already know him.I was about to write to him for some info. but I will wait now to hear from you.
Hope this will be of some help.I may had told you lots that you already know but better twice than not at all.