It is fairly typical for candidates to run out of time when in the run-up to exams in Years 10 and 11. One of the things teachers usually/should practice is working under timed conditions - carefully reading the instructions on the paper, the amount of time to allocate to each question, planning the answers etc. And most candidates do improve greatly over the two year course.
You may realise why the school cannot use your independent assessment report if you refer to the current regulations, section 7.3.5, (page 84).
[to find this, google JCQ access arrangements and click the link at the bottom of the first page to download the .pdf "Access Arrangements and Reasonable Adjustments 2013-2014 (standard PDF version)"]
What has tightened up very much this year is the amount of evidence required - not just low test results - from within the school to complete the pen portrait. See page 87, section 7.6.1, for information required, especially the parts highlighted in yellow.
JCQ is doing formal inspections of schools with regard to access arrangements, and they are inspecting the evidence the schools hold to back up the access arrangements requested. As I understand it, there HAS to be evidence from (at least some of the) teaching staff to support the assertion that "the candidate has cognitive processing difficulties which have a substantial and long term adverse impact on their speed of working". Standard procedure is for the SENCo to contact the candidate's subject teachers for their written observations on the candidate.