What are my chances of getting a refund?

I ordered some things from a website the other day, and I needed them for Saturday and I noticed they had Saturday delivery, so obviously I chose that one and I paid extra for it.
Saturday came and they never turned up. I paid by Paypal so I've opened a dispute stating that I never recieved my items on the day I paid for.

I have emails stating what I paid for and them telling me that my items were shipped.

So what are my chances of getting a refund?

What website were you trying to purchase from?
I'm not sure about refunds but I'm pretty sure you can ask for either a discount or voucher code etc

BooHoo.com is where I ordered from, they offer refunds and I'm just waiting for them to reply to my dispute thing to see what they say.

Hopefully they'll give you your refund seeing as they didn't deliver on time :)

the delivery company didn't deliver on time.

You're supposed to claim with the delivery company, not the store afaik.