Colin, I did eventually get my account approved manually by Twitter for read/write API - without any mobile phone linkage.
However… there was virtually no communication - I too filled in the form (got a fairly prompt robo-acknowledgement) - then after about 3 days I got a response from support@twitter.com saying it should be sorted within 48 hours - if not, reply to them.
I did as instructed and then also chased again about a week later.
Finally, a couple of days after that, but with no email confirmation, the permissions were fixed. Note that you actually have to change from read-only to read/write using the API admin web form, the twitter admins don’t do that for you.
I would fill in that form again if you didn’t get any acknowledgement (check your spam)
HTH!