Trinity, Whitgift and Royall Russell (all in Croydon) test in year 8 for 13+ entry.
City of London Freemen's in Ashtead could also be an option, with a pre-test in year 7 and a test in year 8. This is what they say on their website:
Year 6 and 7 Pre-Test Examinations
Pupils can sit pre-tests for Year 9 entry in the January of their Year 6 or 7. We will send you details of the entrance examinations in the autumn including the number of vacancies in the year group. You will be asked to confirm your child’s entry for the examinations.
On the basis of these tests, pupils are offered a conditional place subject to performance in either Common Entrance for pupils at Prep Schools (average 60% required) or our entrance tests taken in the January of Year 8, for pupils not at a Prep School.
Pre-tests are currently taken in English, Mathematics and Non-verbal reasoning.
Pre-tests are only offered once a year. Late registration means that an applicant will have to wait until the following year. The examination will be held on a Saturday 9 January 2016.
Applying in Year 8
Normally, it is the case that not all places for Year 9 are filled through the Year 6 and 7 pre-tests so it is usually possible to sit an entrance examination in Year 8 for an unconditional offer of a place in Year 9. In the event that Year 9 is full at the time of the examinations, applicants can sit the examinations to obtain a place on the Waiting List.
I would google the Good Schools Guide and search their website for independent schools in your area. You don't have to subscribe to get basic data, such as schools' names, websites, addresses and telephone numbers, so it shouldn't take long to compile a list of suitable schools to contact.