Lateness is an interesting concept. I personally hate being late and get in a bit of a state if I'm late for anything work related.
The last time I was late for work was down to an out of control horse on a country lane. The horse dismounted it's rider and I had no phone signal to let the people know I wasn't going to arrive on time or get help. I had to stop and help the poor lass as she was in a bit of a mess and the horse wasn't having any of my horse whispering technique "Good horse, be a good boy now". Thankfully she got him under control but when I turned up 45 minutes late to the meeting covered in mud an smelling of horse everyone just thought it was funny.
I work with a couple of people who are always late and to be fair I just accept that they are going to be late but one scouser I work with was terrible for making up excuses that weren't plausible. The thing that pissed everyone off about her was we'd spend ages talking about her flooded kitchen or her sick dog episodes in a meeting that was already running late.
Culturally I think in this country we're quite accepting of lateness whereas in other cultures it's not as acceptable. Notably serious about this are the Swedes where the only thing more terrible than turning up late is actually turning up early.
I'd just apologise for being late and let that be it. People will decide individually how they feel about it no matter what drama you invent for your lateness.
If it's a constant work issue though
Orang Utan I'd explain to them the difficulties you have and be up front about it which would be better than them nailing you down for your punctuality and ending up in performance management proceedings. .