Bleaching out stubborn blue dye (advice?)

No. I'm afraid you really do have to leave your hair for at least a month after bleaching, and give it a lot of TLC in the meantime, to get away with multiple bleachings. You'll still damage your hair as damage is impossible to avoid when bleaching by the very nature of the process, but spacing them well apart minimises it.

My advice would be first off to use a colour remover. The best I've used on semi permanent colours is Scott Cornwall's Decolour Remover - you can get it in Boots (make sure you get the remover and not the stripper). Some people have also had good results with Jo Bazz, which I think you can get in Save4rs. The Jo Bazz is a lot cheaper too. You can use a remover up to three times and it doesn't damage your hair - it can dry it though - so I'd say give that a go, then see where you are 🙂

It sounds to me like it might be staining which is very difficult to remove. I would try a color remover before using anymore bleach. You really shouldn't bleach again for at least a month. You also shouldn't use dyes with peroxide as they will further damage the hair. Is there a specific reason you are using a permanent orange instead of a semi? Adore have a wide range of natural and unnatural colors such as oranges which would probably take better on your hair with no more damage.
Orange and blue are opposites on the color wheel so they will tone each other out. The best case scenario will be that the blue is light enough to tone some orange out and those parts will look lighter than the rest.

My only issue is if I don't dye it this week, it will be an entire year before I am able to do so again. (And I don't want half my hair to be greyish greeny blue with blonde, and the top half to be almost blacky brown). 🙁

I'm using permanent orange/ginger as that's the colour I want to be for the next year. It needs to be a permanent shade so it doesn't all fade out like a semi, as with my hair length I can only do root touch-ups when at uni until next summer when I move home again. (Uni is 6+hrs away).

I'll try some of the products suggested, hopefully they lift it some more. I don't think it needs to be super platinum blonde before I dye, I'm just wary of the blue&green tones.

Do you think if I put permanent orange/ginger over this colour, it would go more or less an orange/ginger shade? Or would the blue completely offset the orange tones to a neutral or maybe greenish colour? (If worst comes to worst, I'd rather be mostly ginger/orange with some odd highlights than blueish grey).

Thanks for the advice so far! I will refrain from bleaching any further.

I have never used orange dyes as i don't really like the color but from what i know, it should come out orange. It wont completely neutralise it.

Permanent dyes fade quicker than semi's so this one will go a dull orange color in a few weeks. You could always top up the color with a semi dye every few months. It isnt difficult. I always us to dye my hair at uni. Are you not going to go home for the holidays?

Yeah I used to dye my hair in the uni shared bathrooms all the time. Me and a mate would go in there and do each other's hair too 😀 Plus yes as puerkz says you won't get a permanent orange from a "permanent" orange anyway.

Yup yup another Uni bathroom dyer here, as long as you clean up a bit after it should be fine 🙂

I dyed my hair all throughout uni--like the others said, as long as you clean up after it's not a big deal.

Regarding using a permanent ...

1)  There's still peroxide in there, and fairly strong peroxide.  You're risking fried hair.
2)  A permanent dye over freshly bleached hair is really not very likely to turn out the colour on the box.  For one thing, the hair is more porous, and for another, it's just gone through a process, so it doesn't react like it normally would.

I'm speaking from experience, unfortunately.  Several years back, I bleached my hair and then put a permanent red on top.  My hair was crazy dry and crispy, and it went bright pumpkin orange.

I really wouldn't risk a permanent dye.