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From Newcraighall Road (accessable from both the A1 and Niddrie Mains Road) take the roundabout exit to Edinburgh Fort Retail Park (NOTE: this is a different roundabout from that for Fort Kinnaird Retail Park). Then take the first left at the second roundabout. This road will lead away from the Fort, and turn a sharp left (amidst dumped furniture and white goods at the roadside). After the corner, there will be a right turn, this is the road to Millerhill. Follow it up the hill, and over the top, until you cross the bridge after the farmhouse. Park somewhere here - this is the bridge that goes over the northward access line and Newcraighall turnback siding.

Just at the side of the road - there's plenty of space, and often one or two folk parked there to watch the trains, too.

By Rail:
Train to Musselburgh from Edinburgh.
When you alight, and reach the bridge over the station, turn AWAY from the main town (the way the road goes) and instead go though the pedestrian gate onto the cyclepath. Follow it up to the A1, where it will turn a sharp right, down to the bottom, where you turn left and go under the A1. Up the other side, and you emerge almost at the bridge over the access line and siding. (Turn right and you'll be there).

Newcraighall station is only a few hundred yards away and it is possible to get to Millerhill from the station - it takes about 10 minutes.
From the platform you can see some playing fields to the North East. To get to these you go out of the station, turn right back under the railway and then right again. On the far side of the playing fields there is a footpath/cycle path that goes through a tunnel under the main road and carries on directly to a point a few yards away from the depot entrance. Bus:

Bus to Musselburgh station, and follow directions above.
Bus to Edinburgh Fort, and road directions as above, although the road is not really pedestrian friendly (althought it is very quiet), despite also being a cycle track.

There is a pub near Millerhill Station that does reasonably priced food.

scot-rail.co.uk

There are several points you can get a view of the yard:
(All directions assuming you start off facing the yard from the overbridge)
See the picture descriptions for more details.

Edinburgh Waverley
Prestonpans Station