quick based team around mary?

Up to you, but it's generally not recommended to use 2 riders in the same team since they end up stealing stars from each other. Mary can kinda get around this with her absorption skill, but that only lasts 1 turn.

By the looks of that list, you seem to mostly have DPS, which, if you want a team to be optimal, won't go well with Mary. If you have Hans I highly recommend using him, he is an excellent crit support. If you want extra survivability to your team, then maybe replace Mozart with Jeanne, though she will also steal stars from Mary, so it's a trade-off.

Shuten is also another decent servant here. She has 2 quick cards, so she can quick chain with Mary, and as an assassin, she will generate more stars for Mary than most (there are better assassins for this though. Do you have Fuuma?). Assassin Scathach will serve the same purpose, but deal less damage than Shuten.

Cu will also be fine in any quick team, but he's really relegated to back row most of the time since he has no supportive capabilities at all.

Just saying though, it seems like Mary is unfortunately one of those servants that don't go all that well with most of your best units.

Most players considered that Quick team is not existed before the coming of Skadi or Osakabehime (typically Skadi)

Some servants are Quick oriented like Okitachan, Shishou OG and Kintoki Rider but you will find that Waver/Hans are good general support for them. Okita/Jack/Kintoki are so far best "Quick" DPS due to their common traits: Quick card are in high performance of both star generation and NP gain. Shishou OG even not relying on her poor performance of Quick card.

Mary has amazing ONE-ROUND PUNCH. But generally unlike the three Quick servants I mentioned above, her NP gain and Star generation are quite mediocre, which means before or after that round, Mary is quite struggling for NP charged (AQAe only has ~27.7% NP gain).

So there are several ways to improve heir performance (but seriously, not "Quick" team).
(1) instant charge: Waver, Helena, etc
(2) Star generator, more star for RIder==More critical==More NP gain and damage, etc.
when not facing casters, Jack, Shishou A are suggested due t their affluent quick card, relative low star adsorption and high hit-account could be idea for Mary
when facing Casters, you need equip 2*2030 to Caster supports, and using Hans/Helena.
(3) More stars from Mary? Caeser, using his 3rd skill. The future Caster Gil can also be a boost, but it is really not a Quick team.
(4) You can also have Jeanne in team for better durability and potential Art chain. But still, seriously not Quick team.
(5) Quick-up. Unfortunately Atalante and Shishou's skill only last one round. They can be alternatively choice, but besides of that 1-turn Quick-up, they really cannot provide more help to Mary and team.

All your servants listed, besides of Jeanne and Shishou, I cannot see other servants are in better choice than Hans/Caeser. Riders will fight each other for stars, so Ozy and Ushi are not recommends, even though Mary has 3rd skill, but only lasting for 1 turn.
Billy and Cu are the some. Billy himself are also struggling same problem as Mary, and Cu give almost noting to Mary. team together 1+1<2.
Shuten is somehow recommend, She can help Mary for that One-Punch.
But I dont think that effect could be better than Liz and Waver. Only 2030 Shuten may show advantage
Tamamo for Art team.

I would recommend jeanne, tamamo, and hans with 2030 as support. For star gen, look towards okita and ca hassan. Other assassins are not recommended as if you use anne mary as dps against casters, they will destroy your assassins. Ultimately, you need to run her in a mix of quick/arts comp.

I'm also trying to build a team around A&M after I got her off the Singularity gacha :)

Most people would recommend to get a crit star generator and a support to match with A&M who will be your main damage dealer due to high star absorption. So I made a team with the following:
1. Shishou Assassin (I think Shuten can do as well) for star generation
2. Hans/Waver for support
3. Caesar for the buffs
4. A hard hitting Rider like Ozy as your backup rider in case A&M dies early, you can still take advantage of the stars generated.
5. FSN Cu as my default last man.

Best CE for A&M is Black Grail if you have it as it works with her gimmick of low HP = higher NP damage. Since I don't have Black Grail, I'm trying to run her with a MLB Princess of the White Rose (Guts + 20% NP charge) for survivability.

Best Mystic Code can be either the summer one with the quick up and np charge skills or the Chaldea Combat Uniform for the attack up and the order change allowing for a last minute buff from a unit like Caesar before A&M uses her NP.

Hope this helps and good luck with the team building!