How do I apply for a DVLA provisional license without a British Passport?
Yes, you can obtain your DVLA provisional license without a British Passport, as being a legal work-permit holder and a visa national, you can obtain this asking DVLA provisional license, and you DON’T need any British Passport to obtain it. This is the Rule, as defined in their Immigration Rules HC 395. Secondly, as you described that their online application form does demand the British Passport number, so it might be a soft-ware error and incompetence of the programme, but it does not mean that without British passport, you cannot get your DVLA provisional license. Simply go to the nearest DVLA office and ask for their provisional license or go to the nearest driving training institute, they can arrange so.