I wonder. Obviously there will be an easement (written or unwritten) allowing the neighbour's rainwater runoff from the roof to run along the gutter and discharge into the downpipe on the OP's side.
Assuming the neighbour is not running any new pipework outside of his or her boundary and the drain is a combined foul and surface water drain, then I wonder why the OP would want to refuse permission for the neighbour to also discharge condensate into that same gutter, although it may be the case that since there is no historical use of the gutter and pipe on the OP's side of the boundary for condensate, permission would need to be sought. The OP would do well to obtain legal advice from any free legal advice that may well be provided by his or her home insurance or trade union.
Understandably the OP may not be happy if they wish to use the downpipe for a water butt, if the downpipe is some material other than plastic that might actually be damaged by the condensate. If the pipe runs to a soakaway not specifically designed to take boiler condensate or to a municipal surface water drain then it should not be taking boiler condensate whoever owns it. Finally, if the neighbour is actually placing a condensate pipe on the OP's side of the boundary then that would also, I think, require permission.
The fact that the OP might have grounds to object does not necessarily mean he or she should, but that is another matter with which we are hardly able to pass judgement on.
