An update after more work. This is what I've found, and have also posted on artist John Cheltenham Wake's bio on Wikipedia since it originally said 'not much known about his early life'.
Now I can see where Elizabeth Lewis fits in.
John Wake (abt 1808-1868) married Sarah Cheltenham (1810-1863) on 24 Oct 1831 at St Martin in the Fields, Middlesex, England. They had seven children including John Cheltenham Wake.
John Wake was born in Bloomsbury in about 1808, son of John Wake. Sarah Cheltenham was born in Shaftesbury (not Shrewsbury as in the records - someone must have misheard/misread the place name) in 1810, daughter of John Cheltenham, victualler (of the Swan Inn, Shaftesbury), and Sarah Cheltenham nee Hunt.
John Cheltenham Wake:
Born 27 Sep 1834 and baptised on 19 Oct 1834 at St George Bloomsbury, Camden, Middlesex. Records show John, son of John and Sarah Wake, of Duke Street, father's occupation a coach trimmer.
1841 census: Age 6. With family at 38 Monmouth St, Finsbury, with father's occupation a coach trimmer.
1851 census: Age 16. With family at 7 Rose Square, Islington, Liverpool. John is listed as 16 years old, an apprentice gilder, born Middlesex. With parents (John 43, born Bloomsbury, London, coach trimmer, and Sarah 42 born Shrewsbury [sic]), siblings Ellen, Douglas and Alfred. NB Alfred was their only child to have a middle name of Cheltenham registered at birth.
Later censuses show the parents and rest of the family remaining in Liverpool.
1861 census: John Cm Wake, single, 25, lodger, Artist figure & marine painter. In the Dobell household in Erith, Kent. No addresses on this page, and no number for this household. Possibly Park Spring Terrace or Court or Place. First appearance in the records of the middle name Cheltenham. Wrong age. Should be 26.
Marriage of John Cheltenham Wake to Clara Reeves (1843-1922) on 5 Apr 1862 in Brentford, Middlesex.
Birth of son Herwald Andrew Wake in 1865, Dartford, Kent; baptised on 22 Oct 1871 at Holy Trinity, Blackheath Hill, Greenwich, Kent. Baptism record states born 19 Aug 1865. Son of Cheltenham and Clara Wake. Father an artist. Of Thurston Rd, Lewisham.
1871 census: 47 Thurston Rd, Lewisham. Listed as John Wake, 34, landscape artist, with Clara Wake, both born Middx, London, and son Herwald Wake born Erith, Kent. Wrong age. Should be 36.
Birth of son Cheltenham Wake Lewis (1872-8), Islington, Middlesex. Mother Elizabeth Lewis.
Birth of daughter Katherine Mary Wake Lewis (1875–1909), Kensington, Middlesex. Mother Elizabeth Lewis.
Birth of daughter Beatrice Betty Wake Lewis(1877–1958), Shepherds Bush, Middlesex. Mother Elizabeth Lewis.
Birth of son Cheltenham Wake Lewis(1880–1926), Shepherds Bush, Middlesex. Mother Elizabeth Lewis.
1881 census: 3 Caxton Rd, Hammersmith. John P Lewis [presumably John Cheltenham Wake], 42 [should be 46], head, married, draughtsman, born Middx, Elizabeth Lewis, 40, born Ash, Kent. Kate Lewis, 6, Beatrice Lewis 3, C. Lewis 4 months. Andrew Wake [transcribed Wales], 15, visitor, scholar, born New Cross, Kent. It would seem they went to extraordinary lengths to disguise themselves, with variants on names, occupation, birthplaces, omitting Andrew's first name, and referring to the baby as C, yet he'd already been registered with the name Cheltenham so it wasn't as if they were still deciding what to call him.
Died 1882 age 45. Should be 47. Error in age probably due to informant error and/or ignorance of true age.
Died at 5 Caxton Rd on 18 May 1882. Probate on 5 Jan 1883, London office, stated late of 5 Caxton Rd, Shepherd's Bush and 74 Newman St, Oxford St, Middx. Will proved by Elizabeth Lewis spinster of 1 Lime Grove, Uxbridge Road, Middx. Sole executrix. NB There was no number 5 Caxton Rd in the 1881 census and the will, although handwritten, clearly shows 5, not 3, so not a typo.
And regarding Elizabeth Lewis (my original query) and Clara Wake nee Reeves:
Elizabeth:
1887 still at 1 Lime Grove, Hammersmith/Fulham, Middx - address given for Cheltenham Wake Lewis when he started school.
1911 30 Coverdale Rd, Shepherds Bush, Middx. Widow, letting apartments. With son Jack, 30, married, gauger, and daughter Beatrice, 33, single, no occupation.
1921 12 Boscombe Rd, London W12. Widow, 80yrs 11mths, born Erith, Kent. With daughter Beatrice.
1922 death registered in Hammersmith. Could be her but not verified with a cert.
Clara:
1887 electoral register shows Herewald Wake renting from Mrs C Wake, furnished, 1st floor, back. 5s per week. 30 Delaford Street, Fulham, Middlesex.
1881, 1891, 1901, 1911? Can't find her.
1921 census possibly her or is this the widow of Amos Reeves? Clara Reeves (not Wake), head, widowed, 77 and 2 mths, born Dartford, with niece Ethel Reeves, single, 52 and 10 mths, born Brixton, music teacher on own account. Cannot identify the niece connection.
1922 Clara Reeves death registered in Lambeth. Again could be the widow of Amos Reeves.
So the puzzle of Elizabeth Lewis and the Wake Lewis surname is solved, thanks all. As for Clara, hmm.