You can bring a claim in the employment tribunal if your employer does not deal with your request 'reasonably', wrongly treats your request as withdrawn, or fails to give you a final outcome within three months of your request.
You must launch your claim within three months (less one day) of the date on which your application is treated as withdrawn or the final decision is communicated to you.
The first step, before bringing any tribunal claim, is to contact Acas, by submitting a completed Acas Early Conciliation Notification Form. If you do not take this step, the tribunal won’t be able to hear your claim and it will be dismissed. You must take this step within the three-month time limit.
Acas early conciliation is free. There is more information on the Acas website: Early conciliation. You must submit an Acas Early Conciliation Form in order to bring a tribunal claim, but having taken that first compulsory step, you are not obliged to actively participate in the conciliation process (the same goes for your employer) if you don’t want to. However, disputes over the flexible working process may be well suited to early conciliation.
An alternative to a tribunal claim is the Acas Arbitration Scheme. The results of Acas arbitration are binding. You can find out more about this scheme in the Acas Code of Practice on handling in a reasonable manner requests to work flexibly (PDF, 170kb).
Another way forward for you might be through workplace mediation. Speak to your rep and your employer to see whether mediation might work for you. More information is available in this joint Acas and TUC guide explaining workplace mediation (PDF, 1.36MB).
It is recommended that you take legal advice from a qualified and experienced practitioner before proceeding with any tribunal claim.