Unfortunately the EU/UK doesn't have all of this grouped together in the same way the US does - there are separate regulations on labelling, additives, flavourings, general food law, food hygiene, compositional standards for certain groups of products etc.
To add to that complexity, some of the EU-level stuff is implemented by Directives rather than Regulations. Regulations apply as law in all member states (countries), whereas Directives have to be implemented by local laws in each member state, so there is some potential for differences.
Additionally, just in case that didn't seem like a sufficient challenge to get your head around, since Brexit there have started to be some divergences (small initially, but now increasing) between UK and EU food law...
In general you'll need to ensure that any additives you're using, the quantities in which you're using them, and the purposes for which you're using them, are in accordance with Regulation (EC) 1333/2008: https://eur-lex.euro...elex:32008R1333
You'll see down the left side of this page there are links to various "consolidated versions" - these include updates from subsequent amendments, so the latest of these will show you the original regulation with all of the amendments up to the current date.
As a quick reference you may prefer to use the Additives Database instead - it's much quicker than reading through the entire regulation (although you may need to refer back to that for specific points around interpretation in some cases): https://webgate.ec.e...FAD&auth=SANCAS
N.B. This is one area where there has been divergence between the UK and EU, as the EU has implemented some changes since Brexit that therefore don't apply in the UK. Off the top of my head I don't think any would be likely to affect your product type, but definitely check if unsure.
For any flavourings you're using, the key regulation is (EC) 1334/2008. Flavourings are massively complicated in their own right, so I wouldn't suggest trying to fully understand this area. If you're sourcing these from a decent flavour house they should be able to confirm the appropriate legal name and any restrictions on maximum dosage in your product category for use in the EU.
For labelling under Regulation (EU) 1169/2011, I'd suggest having a look at FoodDrinkEurope's guidance document: https://www.fooddrin...ance-on-FIC.pdf
Reading University also has a pretty good guide (albeit set up like a 1990's website...) on UK labelling: https://www.reading....bel/welcome.htm
There are services that can check your labels for you (for a fee) if needed, and that can provide paid-for advice on other specific areas.
In general you'll need to meet the requirements of Regulation (EC) 178/2002 as you've already found, but also Regulation (EC) 852/2004: https://eur-lex.euro...elex:32004R0852
Additionally, as you're using gelatine you might need to look at Regulation (EC) 853/2004: https://eur-lex.euro...ELEX:32004R0853
Animal derivatives aren't really my areas of expertise, but this component may also have implications on import requirements, so I'd look into that with your logistics people too - better to know in advance than to find out when it arrives at the border 
I'm assuming you're not using anything from e.g. fruits/veg or close derivatives thereof that would then need consideration in terms of pesticide limits?
If you are, then Regulation (EC) 396/2005 would apply. There is a nice database to check maximum residue limits, but that would probably be something you'd need to go through with the relevant suppliers, as these limits generally apply to the base agricultural commodity rather than the finished product.
Similarly there is Regulation (EC) 1881/2006 on contaminants - I'd have a scan through the latest consolidated version of this (linked below) just to check whether any of your raw materials would be in scope, as if so, you'd need to confirm compliance with your suppliers: https://eur-lex.euro...6R1881-20220701
Edit: Might be the longest post I've ever written. Apologies for the essay, but hope some of it is helpful!
