University options for children with dual citizenship, EU passport

My children are all US citizens right now, until we reach naturalised UK citizenship (fingers crossed!)

We're looking at the costs of attending university (private and state) in the US, compared with opportunities for the children if they had the EU national status.

Does someone know if they would be able to say, attend university in Sweden (no tuition fees) as an EU national?  Or has anyone calculated the median cost of attending uni in the UK compared with the US?

Trying to work out what benefits they would have as EU nationals when it comes to schooling.

In the UK and the US, the issue would really be that of residency rather than citizenship.  Non-resident tuition in either place is likely to be really steep.  (I looked into this for myself rather than for my kids.)  I'm afraid I don't have a clue about other EU countries. 

I'm a dual US/UK citizen and despite that I will have to pay full fees when I start uni this September. With staying in accommodation and tuition and everything it comes to about $14,000 a year I'll have to pay myself for uni in the UK. Compare this with $32,000 that private uni would cost in the States. It's all dependent upon residency for tuition in the UK, not sure about other EU countries.

(note: in the below I'm assuming that the kids will be dual UK/US citizens ordinarily resident in the UK.)

here's another factor- if your kids aren't resident in the US but want to go to a US university, they'll either be paying nonresident tuition at a public school or private tuition.  However,  they won't be considered international students because they have US citizenship, so they'll have a much easier time getting scholarships and grants to help pay for it   And of course if they're resident in the UK for the three years immediately prior to beginning university they can pay home fees at a UK uni (definitely a lot less expensive than tuition at a lot of American unis!)  Best of both worlds, I think