I think you are stuck with this, unfortunately. Outside of moving, there isn't much you can do about it. I feel like this is why the hedge is where it is - for their privacy. All you can do is ask them nicely to cut it down half way. Other than that-a nice set of curtains.

we have a similar problem with a beech hedge. when we moved in we ascertained who's hedge it was. after a year or 2 we bit the bullet and asked if they'd mind if we reduced its height and trimmed it back a bit on our side. we agreed on a suitable height and we keep it at that. they are 20yrs older than us and seemed delighted that its basic maintenance is done by us. we would prefer it lower still but they aren't allowing that. sadly you have no right to light but all you can do is ask. they may not be even aware how it affects your view.
invite them over for tea and cake and chat about it.

welcome to GoY too :o)

Now, perhaps wait until the spring when you have become more established and speak to you neighbour and even suggest reducing the height of the hedge and offer to help financially. Otherwise, if there is room, erect a number of tall wooden posts with ajoining 6 foot trellis panels the length of the hedge and paint them with a bright paint and train honeysuckles or some other climbing plants on them.

Who trims the hedge on your side, or does it just get thicker and thicker?

If you don't get any satisfaction from the neighbours when you eventually broach the topic I would contact the Council, probably Planning Dept., for advice.