Not being paid for a training day, advice please!

[KentMum2008]

Long story so I'll try to keep it brief.

My manager told me I had to attend a 'mandatory' training day earlier this month. It was on my day off, involved a trip to London for an exhibition of sorts, and there were several seminars we could choose from to attend (one each) and a few free workshops.
I expressed interest in one particular seminar early last month, and was told in no uncertain terms that I wouldn't be allowed because it was £15 and the owners will only pay for full time staff to go on the seminars. I had to attend a free 'workshop' (which it turned out was totally irrelevant and gave me nothing I could bring back and use in the workplace, a genuine wasted hour)
A colleague of mine was allowed to attend the seminar I expressed interest in. She and I job share (I work 16 hours, she works 18) so I queried why she is classed as full time and I'm not. Apparently it's because she works 3 days and I work 2, although 2 of her days are half days and I work 2 full days!
I mentioned to my manager that it was illegal to discriminate against someone because they work part time, and to not offer an equal training opportunity to me because of my hours. I explained that they could ask me to pay half for the course, as I work half the hours of the full timers, and that is a perfectly legal thing to do, but this fell on deaf ears.
I had all but given up the fight, until I recieved ny payslip this month and saw that I hadn't been paid for the 'mandatory' day either. When I queried this, the owner said she paid for my travel and course so I wouldn't be paid for the day.
Surely this is wrong? For starters I didn't go on a course! And also, they can't expect me to essentially work on my day off for free? My normal working hours are 8-3.30, this particular day I didn't arrive home until 5pm which meant DP turning down a days work to do school runs. He would have earned 3 times what I should have earned in that time, and if I'd have known I wouldn't be paid, I would have refused to go!
That's literally as brief as I can make it. I just need some clarification that they're in the wrong here! I'm having a meeting with the 3 managers/owners this afternoon and I feel like it's an ambush.

[flowery]

Yes they are in the wrong and you should have been paid for working that day, and also should not have been penalised for your part time status. Difficulty is that your colleague is also part time (despite them claiming she isn't).

Would those extra hours unpaid take you below National Living Wage on average by any chance?

[Jenda]

Contact ACAS, they are great for this sort of thing. You should have been paid