Python3, 311255 bytes
def f(A,B):
t,p,T,M=0,[],0,0t=T=M=0;p=[]
while A and B:
p+=[c:=[A,B][t].pop(0)]
M+=1
if c:
l=t;T+=1];h=c>0;T+=h;t=[~t,t][h];M+=1
while 1F:
F=1=h:
for _ inwhile range(c):
if if[]==not(K:=[A=[B,B][not t]A][t]):return T,Mh=0;break
p+=[C:=K.pop(0)];M+=1];M+=1;c-=1
if C:F=0;l=not t;t=not t;c=C;breakF=0;t=~t;c=C;break
if F:[A,B][l]+=p;p=[];t=l;break
else:t=not tB][t]+=p;p=[];h=0
return T,M