If you just had the endlers, I'd say take the risk of changing the gravel when you move. But you also have shrimps, so I'm going to say wait to change the gravel as a mini cycle could harm the shrimps.
If you are going to move the tank first this is the way I'd do it.
Bag the fish, shrimps, plants and filter/filter media as late as you can leave it. Put some water into another bag and leave it open for now. Put as much remaining tank water in the bin as you can without sucking up any muck from the gravel. Follow Chucklett's idea for washing the gravel, then put it into the last bag and tie it. Throw away the mucky water left in the tank and in the water change bucket. Move. When you get to the other end, put the gravel, fish, shrimps, plants and filter into the bin together with the heater. (If the water has cooled too much during the journey put the heater in first and wait till it's warmer before moving the fish). If you think it is possible, I think I would roll down the bag with the gravel so you expose as much as possible without it all spilling out and leave it like that in the bin as it'll make it much easier getting it out of the bin before the fish. Or even put the gravel in a plastic tub in the bottom of the bin. You need to allow the bacteria on the gravel assess to oxygen and ammonia/nitrite so you can't leave the gravel in a tied up bag for a couple of days. Once you've got the furniture sorted, put the tank where it's going, then put the gravel in the bottom. Part fill the tank from the water in the bin, move the fish, shrimps and plants into the tank, empty the bin and top up with new water. And the filter and heater.
Don't forget to unplug the heater half an hour before taking it out of the water.
You will lose some bacteria. The inside of the tank itself will have some, and if that dries, they will die. Similarly the filter casing if the whole thing won't fit in a bag and you have to take the media out. The tank has been cycled quite a few months now, so the biofilm, with the bacteria in it, should be well embedded into the filter media so they should come though with little damage.
Feed the fish very lightly for a few days after the move and keep an eye on your water.
I would also be inclined to do several very small frequent water changes (even if you don't need to because of a mini cycle) just to get your fish acclimated to Southampton water.
Once you know the bacteria have settled down completely and the fish are back up their normal feed, that's when I'd change the gravel. It is going to mean more messing about with the tank, but I'd rather do that than risk the shrimps.
When you do get around to changing to sand, the best advice I can give you is wash the sand very thoroughly. It usually had a huge amount of dust in it. At least with an Edge, you won't have too much to clean