think I see John William in 1911 - stating born Bridgnorth - a Compositer
Interesting stuff .... my money is on Eliza NOT being a daur of William & Sarah tho ...... apart from that Birth Index entry in 1865, she doesnt figure in BMDs or the census*, so, as you say only JW's birth Cert Id's her as Eliza Hartshorn(e) - and that wont show her age .....
The tangled webs ......
*Update, may have found her as an actual Hartshorn in 1901, Ashby de la Zouch - Leics. Says born c1870, Shrewbury - a Boarder with the Ward Family - occupation - Laundress.
No sign of young JW tho .....