I think he'd be like Rakitic or Andre Gomes there. Decent but not a perfect fit. Rakitic was playing as a 10 or support striker his final year at Sevilla, and at Barca his job is to play as a CM on the right, filling the right wing spot when Messi comes inside but normally playing as a normal CM in a 4-3-3. Alli would probably be more dangerous making late runs into the box than Rakitic is, but not as good an overall midfielder.
Gomes is a smoother player so you figure Alli would give the ball away a bit more but I think he would definitely convert more chances and make more goals. He could also give them an aerial threat against packed in defences which they don't currently have. I don't think it would work perfectly, but it's not inconceivable.
For England, it would help, since the seemingly logical lineup for them going forward is probably Kane up top, Sterling and some wide player opposite, Dele Alli and Lallana in a Barcelona shaped midfield triangle with a holding player to allow them to press opponents and get into the box (Dier or Henderson or some other mediocre option). I guess they might try 3 at the back and it should suit Dele Alli just fine but I don't think that suits Lallana at all.
Of the CL contenders, I think Atletico is the one who would suit him perfectly. He reminds me a ton of Raul Garcia, who was very productive under Simeone, but Alli is a better player already despite being much younger. I don't think Atletico would pay the money Spurs would rightfully demand for him, though. If Pochettino stays, he should probably just stay