You can find out on your Motability Scheme online account if your application has been approved.
You can contact MINI through Facebook messenger or online chat on their website mini.co.uk and give them your order number (your dealer will be able to give you this if you do no know it). They may ask you a couple of questions to check (like colour or spec), and they’ll give you your order status and (provisional) build week.
Quicker than asking dealer as they will have to contact MINI and get back to you. I got my build week a few days after I ordered my car. A couple of weeks before it was due I contacted MINI on FB messenger to check, and they told me it would be another 3 1/2 months. Next day I got an email from my dealer telling me the same (complete coincidence); so I knew before the dealer by going straight to MINI.
Current lease: 2025 Omoda 5 Estate 1.6 TGDI [147] Noble 5dr 7DCT
Last lease: 2022 MINI Hatch 2.0 Cooper S Classic Premium 5DR Auto
First lease: 2018 SEAT Leon Sport Tourer 1.8 TSI FR Technology DSG