I used to work in a customer service role,where we started at 8am prompt. That's when the call lines opened. I drove and it was approx 25 miles away form where I l lived, up the M1. Lateness was extremely frowned upon so I always left at 6.00am arrived early, sat in the car pack at work,with a takeaway coffee that I picked up en route, from 7am waiting for the office to open at 7.30am.
Yes, I was up at crack of dawn but I was always there on time. I saw others dismissed for their regular lateness ( we had to be ready to receive calls at 8am). Some companies and some roles are time specific so you have to be there. Bums on seats. I always allowed myself enough time to get where I needed to be. If I wasn't there it would have meant my colleagues would have had to pick all my calls . Equally, I would have been sacked and I needed the job. Lateness is one of those things where people can be perceived as lazy and/or sticky mattress syndrome. Once in a blue moon, if the beast from the east unleashes a snow storm from hell, then fair enough. If you are relying on buses, you may need to get to the bus stop significantly earlier, allowing yourself time for delays, breakdowns, adverse weather etc. Teachers, fireman, police and ambulance drivers all need to arrive on time. You need to wake up earlier OP and allow more time at the bus stop. Get the earlier bus or if that is not running a combination of walking and the nearest
In service buses that are running at that time to arrive at work on time. Grab a coffee when you arrive at work early and do some family admin while your waiting to start to work.
If this job is difficult for you to arrive at punctually, you may be best off finding something within walking distance of home. What your bosses originally said they would accept is clearly not the reality so you need to either get there in time or find a diff job. It could be that they use this to dismiss you in the future so you need to ensure that you give them no cause for concern. If you need your job in order to live you need to arrive on time.