IANAL and you should speak to one but my reading of this:
Obviously a lot of it depends on the contracts between the three main parties and the country you're in but I suspect it's fairly straight forward (and probably not what you want to hear).
In terms of the relationship between you and the supplier (that is the company who employed you), you need to look at how you were employed and how the project ran.
If anyone else did any work on that site at all (including relationship management, analysis, copy writing, testing or whatever) then legally it may be that you did not create the site, you were part of a team creating the site at which point the ownership would rest with the umbrella under which the team were operating (the company). I'm guessing that this was the most likely situation.
You also talk about back pay rather than invoices so I'm assuming that you were contracted on a daily rate / hourly rate basis rather than working on a specified project basis. You're in the UK so IR35 might help as an indicator here - do you pay PAYE? If so then it's far more likely that you'd be seen as an employee and have no rights to the IP.
If my assumptions here are true, I don't see you have much of a claim.
Even if they're not it's not a given that the IP will have reverted to you. They have breached the terms of the contract but that doesn't necessarily void it, it just means that you have a claim against them.
The client on the other hand had an agreement with the supplier to supply a website - they paid for it and received it so would seem to have a pretty solid and reasonable expectation that they own the IP.
If you think that you do have a claim on it after that then you need to think about what you're actually going after the "client" for. They're not your client, you have no contract with them (express or implied) which means that you'd have to go after them based on infringing your IP.
That is likely to be far less straight forward - particularly as they have a reasonable case that they own the IP which is likely to mean the whole thing will likely cost more than you can ever recoup.
So all in all I suspect you're better hammering away at the company you were contracted to and leaving the client alone. It's not necessarily a case of do you have a claim, but where do you have the best claim, and that would seem to be for straight forward breach of contract.
That said, as an aside, have you thought about approaching the client and saying "BTW, I was the guy who actually did that work, would you prefer to contract direct with me in the future and save money?". That would potentially seem to be a win win.