Identification and advice please

Cheerful-Juniper

Good morning - we moved into a new house last September and are still finding our feet. The garden is beautiful and very mature and a great deal has obviously gone into it. There's only one fly in the ointment, the humungous hedge that borders the east and north borders. The east side is onto a road so is actually very useful, blocking out noise and giving us privacy, the North side is our neighbours front garden, the house has been empty since we moved in and the executes haven't been in a hurry to sell, obviously the corona virus has halted things completely although I'm not convinced it made any difference tbh.

Anyway, picture this has identified the hedge as lawsons cypress, what's the difference between that and leylandii? I want it down, it's taking up about 10' of garden and it must be severely reducing the light into our neighbours garden, although not an issue atm. The title deeds also state any hedge, fence etc can't be larger than 6', these must be about 12'.

So any words of wisdom are much appreciated, I've uploaded photos from my side and also from my neighbours side. 

Fairygirl

It's slightly irrelevant - if you don't want it, remove it! 
It might be possible to have that done depending on what local tree surgeons/landscapers etc you have, but just be aware that it's not the best time of year, as there could be birds nesting in it. It's illegal to disturb them.
Generally it's a 2 metre height fro boundary hedging, so yes, it would be worth investigating how neighbours feel about you removing or reducing it. If you cut back into them too far, they don't recover. These varieties still  need careful attention from an early stage to keep them at a good size. 

The Lawson cypress is a Chamaecyparis, and  has a denser habit than  the Leyland cypress [leylandii] and the foliage is slightly different. It tends to be more shapely too.  If you look on the RHS site or something similar, you'll get info about the different foliage and habit. 

Cheerful-Juniper

Yes, we'll definitely wait until autumn. Will it be possible to save the hydrangeas and plant them elsewhere? 

Fairygirl

Sorry - I just saw this.
The Hydrangeas might be tricky to move, especially at this time of year - better in autumn too. They look very nature and established. 
You could take some cuttings from them, but of course, they'll take a long time to become big shrubs.
If you do want to move them, follow the usual rules of soaking thoroughly first, getting as big a rootball as possible, and water thoroughly after until established. Prepping the new site first is also useful, as it means minimum stress for the plant. 
Cutting them back also helps to get them established more readily, although it will mean losing flowers, but that's a small price to pay. They're tough, so it's worth having a go 

Cheerful-Juniper

My husband is having none of it. I've persuaded him to reduce the height of it but that's it. I'll have to wait until we have new neighbours to push for removal 😂

We also have this. Beautiful to look at but starting to push out the wall u derneath, plus there's a beautiful magnolia tree behind we can't see at all unless we venture into the forest. The garden is stunning and one of the main attractions of the house but it's just starting to dawn how much needs done.