To have still gone out yesterday?

It's ridiculous that so many people would have upset their own mums because some man cannot mind his own kids for a day!
And the level of entitlement from some men!

You've done absolutely nothing wrong, I would have done exactly the same. You had plans to have a lovely afternoon with your DM and DC which your DH was fine about (ofcourse why wouldn't he be). The DSC having to come because their DM decided to work on Mother's Day rather than spend it with her own DC does not mean you should change your plans.
It's likely they would have rather spent the day with their DM. The who dynamic of the meal would change from mothers and daughters if you start inviting DH and DSC. I can't believe your DH was so selfish to try and make you feel guilty on what is really your day.
Also I don't think you needed to go straight home after the meal, how lovely you could spend longer with your DM on Mother's day. Surely your DH is capable of looking after his DC alone for any amount of time. If not then that's the problem here.
I hope you, your DM and DC had a wonderful Mother's Day 💐

Where did I say their DC should stay home?

But his kids should quite easily have understood that you already had plans and it was their plans that changed last minute.

When else in the year do we celebrate "Mothering Sunday"?

If the kid "dwells" on their step mother taking her mum out for Mother's Day to the point it would "ruin their relationship" then that is pretty unreasonable and bratty and may be a sign that they need some help to get a psychologically healthy perspective - otherwise life is going to be quite a horrible experience for them!

Maybe the OPs Mum getting her Mother's Day plans ditched and imposed upon in favour of a step child would "ruin their relationship"?

So what about her DC spending time with their mother? It was a special treat arranged for the 3 of them so why should they stay home?

As others have pointed out, so why couldn't the child's actual father whisk them "off for a lovely lunch" or any other activity of their choosing for some quality time together, which they would have almost definitely have preferred rather than tagging along to see their Stepmother's mum.

The only issue I see is that your child was going along, so then in a blended family taking his kids on and all that. But his kids (depending on how old they are) should quite easily have understood that you already had plans and it was their plans that changed last minute.

Absolutely no way should your plans have changed, but I think that would be much easier to say if it was just you and your Mum without your DC.

@sleepyscientist You would call your mum and tell her that because your partners ex-partner is working overtime, you're no longer treating her out to a meal on Mothers day and you'll have to rearrange to another day?

Your poor mum!!

1. It's MOTHER's Day, not Children or Step Children's Day.
2. OP had plans with HER mother and child/mother's grandchild that her DH unfairly expected her to cancel at short notice. It's nearly impossible to get a table for Mother's Day at the last minute. I tried and failed yesterday, in 4 different towns in a 15 mile radius.
3. If anyone is responsible for the child's hurt feelings it's their OWN mother. She's the one who prioritsed overtime over her child on Mother's Day and imposed on OP when she already had plans.
4. "D"H couldn't be arsed to do anything nice for Mother's Day until his other child gor dumped on them. He had no intention of going out for a meal with his wife, other child and m-i-l. Then all of a sudden he wanted existing plans cancelled and new plans made to make it special for his first child. Does his second child not matter enough to make an effort for? Is it only stepkids whose emotions we need to protect?
5. He would have been doing bugger all had his kid not been dumped on them. It was up to him to do something nice to make up for his ex dumping the child on Mother's Day. It's not up to OP to make up for two shitty parents.
6. Stepmothers have feelings too believe it or not. They shouldn't be expected to be martyrs who prioritise their stepkids over other family members no matter what unrealistic posters on MN are determined to say.
7. Hope you had a nice time with your Mum OP - you and she deserved it. Don't feel one bit guilty.

There was nothing said about DSC’s feelings but their father could have said yes to having his child for the day but for the child and their mum to understand OP and her child wouldn’t be there. OP is entitled to carry on with an arrangement previously made and arranging the lunch for another day is missing the whole point of mother’s day. She is also entitled to go back to her mum’s house afterwards too.

This does seem to be the case doesn't it!

So you would have OP disappoint her own child and mother? That’s pretty mean.

The OP hasn’t blamed the mum for working but this is in actual effect the mum’s issue. It’s her time with dc, and she chose to work. We don’t know why she chose to work overtime on Mother’s Day, maybe money is tight, maybe she has a demanding employer, maybe she did a colleague a favour but this was her time with dc and whilst it’s great that OP’s DH accommodated the change in plans and had his dc last minute, him trying to co-opt OP into caring for his dc is unacceptable.

I dunno I would always put a kids feelings over a meal out, I can always rebook the meal to another day. You don't know if the kid will dwell on it and ruin your relationship with them.

Yeah I thought that. Why didn't he enjoy the quality time with his own child?

No. It means dh has to change his plans. Not you. If your dh thinks you do too, that's on him.

These were my thoughts.

It's a hard one with blended families.

YANBU.

You're not their parent, it's not your job to change your plans last minute.

He's just pissed because you didn't look after them Z

Their step-mum’s mother!

Seriously, what child is deeply disappointed to spend Mother’s Day at home with their dad, instead of having lunch with somebody else’s gran? No child ever.

He wasn’t interested in coming when he had the entire day to himself to do what he liked.
Then suddenly had to entertain a kid by himself for the afternoon, and what do you know! He wants to come and be involved in your plans which he had NO interest in before.
YANBU.

This depends completely on his reasoning which only you will know op. Everyone on this thread hasn't got enough info and is projecting from their own experience from either way.

Either.

1. Does he always expect you to parent his kids. Is that what's about? If so, then he joins the many men who actually don't want partnerships, they just want sex and childcare, and don't care particularly which female does it.

Or

1. Is he actually a good guy, who was thinking about his children missing out on Mother's Day, because their own mother had to work, so he wanted to ensure they weren't sad about it.

Let's face it, it's probably 1.

But also op, you have twice blamed their mother for working. I expect your partner is trying to deflect to her too. But it's not her. It's entirely him. Assuming 1.

Your DH could have easily had a nice afternoon with his child watching films and ordering pizza, but he preferred to try and make you feel bad.
Yanbu.