The first thing to do is to work out which fence/s, if any, are yours.
If you have a look at the title register plan for your house it will often show which fences you are supposed to maintain.
If you don't have this then you can get a copy online for £3 from here:
https://www.gov.uk/search-property-information-land-registry
The title plan often shows a "T" symbol on the side of the boundary for those fences that you are responsible for. Effectively, these are "your" fences. Or the Title Register itself (you will need to pay another £3 for this) may also say which fences are your responsibility.
Sometimes there is no indication on the plan or register as to who is responsible for a particular fence. In that case it is usual (but not always) for the posts to be on the inside of the fence that you are responsible for and that the "good" side of the fence faces outwards.
This isn't always the case though and there is no rule that it must be done.
Looking at your photo it appears that the fence in the background also has the posts on the same side. I would guess that all the houses here have the posts on the same side. You will likely find in the title plan that it shows that you are responsible for the fence on one side and your neighbour is responsible for the fence on the other side.
Most likely, the fence you show in the photo is "your" fence as it has the posts on your side, but do check with the title plan.
Since this is your fence then you can do what you like with it. The maximum height for a fence is 2m (or 6' 6").
If you plan to do anything major to the fence then it might be a good idea to just let your neighbour on the other side know what you are going to do.
Since it's your fence then you can do anything you want with it. If you want to replace it with an entirely solid fence that is 2m tall then you can do that. Or you can simply board up the gaps yourself and stick some trellis on the top if you prefer.
"I was thinking of getting 5 ft high fence boards to fill the gaps in with and then installing a trellis on the top."
It's your fence, you can do exactly what you like (within the 2m height)
"... and the fence opposite doesn't have the fence panels on my side, it's on the neighbours side."
In this case, (assuming your neighbour is responsible for that side) you can't attach anything to the fence without their permission. The simplest thing would be to ask them first.
If they didn't agree then you would effectively have to build your fence on your side. So put up your own posts and fence panels, these can be totally solid and up to 2m high.