Any question answered...

I was given a pair of shoes for xmas in the wrong size, they are are completely unused but the store refuses to change them for the right size because I don't have the receipt - do they have a legal obligation to exchange them or am I screwed without the receipt?

Even if they did have a legal obligation the onus would still be on you to prove they were bought there in the first place.

Does the buyer have any proof of purchase - that could include a card statement.

Have you tried talking to the store manager nicely / going in when different staff are on duty?

As a last resort, find somewhere else that stocks them and see if you can swap them there