But... They don't find out about your state of health untill AFTER the offer has been made...

Which means... They don't factor that in to whether you will receive an offer or not.

Once you have been offered a place...
And it is signed and sealed...

Then done done can't be undone.

At least... Not without getting in major trouble with current legislation regarding discrimination etc etc.

I really think that you will be okay.

Sounds to me like...

They will offer you a place or not on your merits (Good Luck!!!)

And once you have a place....

Then they are concerned to look out for you.
Look out for their 'investment'
Which is kinda nice if you think on it