Maybe this is a bad time to point this out, but...
Tweed amps are built with the heaters wired up first, before any other elements of the amp. This is where seeing inside an original really helps.
I use to own a couple of original tweed Fender amps. Space inside is very limited (as you've found out).
The power transformer, power switch, pilot light and tube sockets get mounted first, and the heaters are wired. As you've probably noticed, there is a small lip around the edge of the open side of the chassis. The twisted heater wires run along the small corner formed at the inside of this lip, with wires splitting off from this line and heading over to the sockets themselves.
The idea is that this keeps the heater wiring against the chassis and as far from the circuit board and signal wiring as possible. This is also backwards of how you might see blackface amps wired. That's due to the fact that the sockets and the circuit board are on the same plane (as opposed the the 90-degree rotation seen in a tweed chassis).
The next thing I would wire up is the front panel pots and input jacks. The last step for me is to install the board itself. You generally have to insert it starting with one edge of the board, then rotate the board to lay it in flat. Keeping the wires running from the board to the pots and sockets oversize is very helpful, until you are ready to solder them in place.
The board wiresgoing to the sockets generally lay over the heater wires at right angles (as much as possible). Wires running into tube grids often sit up in the air to stay away from heater wiring. The wires running from the board to the pots are often dressed against the chassis and run up to the lip of the chassis on the pot side, before making a turn to run down to the pot lugs. A picture is worth a thousand words, but the amps I've built are unfortunately in another state right now.
This methoid of building/wiring winds up looking pretty neat, and gives the illusion of space in the chassis. If you need to swap a resistor or cap, it's quite easy to get your iron to the part. It does however result in some headaches initially for the guy wiring it up, although it's not too bad once you get the hang of it.
For all this stuff, solid core wire is almost essential, as you need the wire to stay where you place it. The real cloth covered wire (like Doug sells) makes cutting the wire to length much easier.