Python3, 294 bytes:

y=lambda j,n,m:{'v':(n-1,m),'>':(n,m+1),'<':(n,m-1),'^':(n+1,m)}[j]
f=lambda p,s=(0,0),c=[(0,0)],l=[]:0 if not p else(k:=((r:=y(p[0],*s))in c))+(min(f(j+p[1:],[r,s][k],c+[[s],[]][k],l+[p[0]])for j in g) if (g:=[[''],[u for u in'v<>^'if y(u,*s)not in c]][k]) else f(l.pop()+p[1:],c[-1],[r]+c,l))

A very unelegant solution, but not entirely brute force, with basic backgracking.

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Python3, 291 bytes:

y=lambda j,n,m:{'v':(n-1,m),'>':(n,m+1),'<':(n,m-1),'^':(n+1,m)}[j]
f=lambda p,s=(0,0),c=[(0,0)],l=[]:0 if not p else(k:=((r:=y(p[0],*s))in c))+(min(f(j+p[1:],[r,s][k],c+[[s],[]][k],l+[p[0]])for j in g)if(g:=[[''],[u for u in'v<>^'if y(u,*s)not in c]][k])else f(l.pop()+p[1:],c[-1],[r]+c,l))

A rather unelegant solution, but non-brute force, with basic backtracking.

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Python3, 231 bytes:

y=lambda j,n,m:{'v':(n-1,m),'>':(n,m+1),'<':(n,m-1),'^':(n+1,m)}[j]
f=lambda p,s=(0,0),c=[(0,0)]:0 if not p else(k:=((r:=y(p[0],*s))in c))+min(f(j+p[1:],[r,s][k],c+[[s],[]][k])for j in[[''],[u for u in'v<>^'if y(u,*s)not in c]][k])

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Python3, 294 bytes:

y=lambda j,n,m:{'v':(n-1,m),'>':(n,m+1),'<':(n,m-1),'^':(n+1,m)}[j]
f=lambda p,s=(0,0),c=[(0,0)],l=[]:0 if not p else(k:=((r:=y(p[0],*s))in c))+(min(f(j+p[1:],[r,s][k],c+[[s],[]][k],l+[p[0]])for j in g) if (g:=[[''],[u for u in'v<>^'if y(u,*s)not in c]][k]) else f(l.pop()+p[1:],c[-1],[r]+c,l))

A very unelegant solution, but not entirely brute force, with basic backgracking.

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