For A levels there are only a few choices, Jumeirah College, Wellington international on SZR, who also do IB, Dubai British school (DBS), English College, Dubai College, Dubai English Speaking College (DESC).
For sure there is a wait list for JC and DC, in fact it wouldn't surprise me if DC don't even take names for 6th form. It is highly selective and if your DS only has soso grades they will not even consider him. JC requires 8 gcse's at C or above with a B for the subjects you wish to study with an A for subjects you might not have studied before. I'm not sure about the others in such detail but I do know that DBS has a long wait list, likewise DESC although I am not sure of their exact requirements.
Something to bear in mind, Dubai kids have a a different kind of freedom than the UK, there are big weekend parties with the "milkman" delivering alcohol and other stuff to kids anywhere they want, including beaches, parks, tower block car parks, the cost is huge and the ramifications very serious, alcohol is illegal under 21 and drugs are full stop. The Consulate visit all the British schools every year to state specifically they do not have a get out of jail free card for drunken drugged up teenagers.
It's very easy for kids to get swept up in the lifestyle, kids have drivers to bring them back from parties, you see all the cars lined up waiting. Otherwise they get cabs, most kids get cabs by themselves from about 13/14 as buses and the metro don't go everywhere so it relying on parents rather than doing it themselves. The money some kids have is unbelievable, I have never seen a 1000dhs note but a boy in my sons class often has one or two of them in his wallet. A number of ds friends have personal bodyguards (fathers have various consulate/government positions), it's different and very easy to get swept up with it all but then they coming crashing back to earth for reasons as simple as some concerts require you to be of 21 because alcohol is sold.
A number of older teenagers live virtually by themselves, parents traveling for business etc, without a support network it can be by lonely for them leading to distractions. Kids can drive at 18 although I would want my child to learn in the uk as the standards here are appalling, and many do drive, cars and petrol are cheap but apparently so is human life to many drivers.
On the other hand it can be a great lifestyle, sailing, beaches, skiing, easy travel to Asia, good weather for 8 months of the year, it's about 30 today with clear blue skies.
Feel free to ask more questions but personally I wouldn't see moving here as a really viable solution, also an increasing number of kids get sent back for 6th form to try for home status for uni (something else to think about, you have to be resident in the uk for 3 years prior to attending uni to qualify for home status).
HTH. and good luck finding a solution.