There are some grammar schools with no catchment areas - if you search the forum you should be able to find previous threads on the subject. Eg. St Olave's in Orpington (boys school though). These schools are super-selectives, meaning they offer places to the candidates who get the top 116 scores (for St O's, other schools may have more or less places). Obviously, competition is fierce for these schools. Other grammar schools have different rules and you really have to look at individual schools and areas to find out how places are allocated. Newstead Wood (girls GS) in Orpington has a 9 mile catchment, for example.
If you're moving to the Lewisham area (which isn't that far from me in Bromley), you may be close enough for the Bexley Grammars (Bexley, Chislehurst and Sidcup and Townley for Girls). If you look on Bexley council website, they have a "transferring to secondary schools" booklet which gives the distances the GSs have gone out to in previous years. With these schools, you need to be "deemed selective" to get in and live within the catchment. That said, the children with the top 180 scores are guaranteed a GS place at the Bexley GS of their choice.
The whole process is very confusing, particularly as it varies from area to area and school to school
I hope this helps. Good luck