Urgent help needed - waterlogged garden

You need drains. If they didn't work for your neighbour that is because they were not built probably, or the soakaway is not sufficiently large.

Big conifers (and a range of other big plants) will do a fine job, but they will take several years to get to the point where they are big enough to do that job (and if the ground is very waterlogged they may struggle to get establised [depending on variety - Willow would probably be fine, but you need an evergreen to drink water in Winter])

If the water is running/coming down the hill, and your neighbour is agreeable, you could dig a "ditch" across the slope (I'm assuming that would have to be on your neighbour's side of the fence) to do the job that the conifers used to do - but the ditch would need to have somewhere for the outfall to flow to. That could be a temporary solution whilst you then do something more long term. It is your neighbour's removal of the conifers that has caused the problem; I don't suppose you have any redress (or maybe it does ??), but that might make them amenable to trying to help you get the problem fixed. Putting the "drain" where the conifers used to be might be the long term solution.

 

The answer is no IMHO.
You can do nothing about the water as its coming from outside of your boundry. Coniffers will suck up a certain amount but they need to be a good size before it makes a difference. Planting bog plants or moisture loving plants will look good BUT will not suck up the water and not get rid of the problem. Spiking the lawn will make no difference as the water content by the looks of it is massive. The only way is to drain it away.
You say you are on a slope with the neighbours house higher than yours. Why isnt the water draining then back to your house which must be lower ?
It maybe the case that this area is in a slight compacted "dip", small dug channel mixed with hardcore may just releive it and disperse the water.
Drains sounds drastic but they arnt really. Im just going through the process of doing mine at home. I have a similier problem. If you have knowhere for the water to go, drain wise, then dig a soak away in a part of the garden close by that is ok drainage wise. A soakaway is a deep pit hole, fill it with hardcore, then run a trench from that pit to the wet area. Line it with a garden membrane, fill it with shingle, cover with the membrance and replace the turf. There are also land pipes you can buy but this would cost more. To make it even better, just run a few trenches off the main one, line and fill in the same way, and this will help to catch more water.
It sounds drastic but not really.

G