It all depends on your contract. Unless you have specifically requested it for the contract, you probably don't have any rights to the original footage (artist/creative workers generally don't like to give out raw footage, unedited photos et c.). Because of this, you will in general have to trust the artist/editor to do a good job for the final product. This is why you should always check their creative portfolio before you hire them to see if you like their style, especially if it's about something important that can't be repeated (like your wedding).
They have used the wrong songs, missed key parts of speeches etc... Their contract states any problems with the final edit should be notified within 7 days of receiving the completed video otherwise they presume that client is happy with the final edit.
If this is the case, you have fulfilled your part of the contract and can refer to it if they later refuse to do the requested edits. But just because you are contractually obliged to inform them about any changes you would like them to make, it doesn't mean they have to answer within the same timeframe. Again, check your contract for what it says regarding their response time. If it specifies a timeframe for their response/delivery of the requested edits, wait at least until that deadline has passed before you contact them again. If it doesn't, maybe try some other means of communication (email/call/personal visit).
What should be my next steps, I do not mind going to court but that seems like a very long winded and possibly expensive route.
Suing should be your last, last resort. If you still want this company to edit your movie (with the changes you requested), it won't do you a whole lot of good to drag them to court. You have hired them to do creative work, after all, and I don't believe they will be very passionate about getting you the movie you wanted after you sued them. I would only consider suing if they straight out refuse to comply by their contractual obligations. Keep in mind that you can only sue them for what was agreed upon in the contract. Getting the source files might not have been part of that, so your legal footing in this case will be shaky at best.
I am considering asking them for their original source footage and asking another company to complete the final edit for me - what are you thoughts on this, what would be the best approach with all of this?
This again depends on your contract, specifically the scope/amount of changes that you can request. Based on your question, it sounds like rather then having some minor issues with the movie (e.g. wrong format, some uncomely shots, bad audio mix or something like that), you don't like the style of the overall movie, which means they would basically have to redo the entire movie. Again I can only guess, but this is probably not covered by your contract. In this case, it might be easier to have them give you the source footage so you can hire someone else to cut your movie for you, since even if they do the requested changes, it still won't be 'your' perfect movie. If you wanna go down this route, your best shot is to be honest and try to have them see your site of the problem. It's a win-win after all (you get the source footage, they don't have to put any more work into the movie).
If they refuse to hand out the source footage or you want to hold them up to their contractual obligations (i.e. doing the edits you want), be polite, give them time to answer, and only threaten them with a lawsuit after you've really exhausted all other options. A lawsuit may get you what you want, but it will cost you way more money and time.
If they refuse to do the requested edits or can't do it to the extent/in the way you want it, you can maybe strike a deal (instead of having them go over several improvement cycles, they can just give you the source footage and be done with it). You can try to get that deal in the courtroom, but it will be easier, less time- and money-consuming for both parties to do it extrajudicial.
