Depending on where exactly OP lives, can try train from Clapham Junction to Cheam for Nonsuch or to Wallington (coupled with short walk or bus ride) for Wallington High. Both schools have 185 open places between them.
Yes, those journeys would be fine; I was only replying to the OP's question about a possibly tricky journey to Henrietta Barnett school (and I have no idea what that journey would involve, so went entirely on what the OP mentioned).
As for Wallington High School for Girls, looking at the map of their official catchment area on the school's website, I think a chance of getting in from Clapham is very slim; their catchment reaches only up to Mitcham/Wimbledon so Clapham is likely to be too far. Nevertheless, it would not do any harm to ring the school and ask if they have ever had any girls coming in from outside of the official catchment. The map of the catchment area is here: http://www.wallingtongirls.sutton.sch.uk/Catchment
Regarding Nonsuch, they offer up to 85 places to the highest-scoring candidates irrespective of their home address and the rest is a mix of a catchment area with a 3.28m/5.25km radius and high scorers who pay council tax to Sutton. See Admissions 2017 document here: http://www.nonsuchschool.org/uploads/as ... 7.docx.pdf
The distance from the OP's SW18 to Nonsuch if probably around 8 miles, so the OP's DD would be able to get in if her exam result ranked in the top 85 girls. NB. I assume the girls commuting to Nonsuch from Hounslow and surrounding areas must have been the top-tier scorers.
According to Sutton's secondary school transfer booklet, last year Nonsuch and Wallington Girl's had 210 places each
, so 420 places available in total. Last year's booklet with info about all Sutton's secondary schools can be downloaded here: https://www.sutton.gov.uk/downloads/fil ... oklet_2016