I got the oscilloscope repaired a while ago. The main problem ended up being blown input transistors, which are luckily socketed (A1A1Q1 for channels A and B on the attenuator boards). The hp replacement part number is 5080-9691, and includes two transistors as a matched pair. I bought a set from ebay, and they're still available, although there's only one set left as of right now:
http://www.ebay.com/itm/300867847783?_trksid=p2060353.m1438.l2649The repair: Before buying the replacement transistors I tried substituting JFETs I had on hand to see if it would fix the problem. Unfortunately the only ones I had on hand were the J111, which have completely different characteristics, and don't bias correctly in the circuit. I added a couple of zener diodes to get the bias point correct enough to kind of work, albeit with limited voltage swing:
[1a]

[1b]

Finally an image on the screen! (although not triggering properly):
[2]

I ordered a replacement set of JFETs, part number 5080-9691. Unfortunately, even cross-referencing the part number, I don't think these JFETs are produced anymore. On top are the original (burnt out) matched pair, and on the bottom are their replacements. It looks like hp changed suppliers sometime during production:
[3]

I took some measurements of the new JFETs for Vgs vs Id. The blue squares and orange diamonds show the measured replacement matched pair, and the other four colors/shapes show four samples of a J310 which I ordered later:
[jfet_curves]

After doing these tests, I think a J309 would have actually been a closer match to the hp parts. And luckily BG Micro still has both the J309 and J310 in stock as through-hole parts! The pinout is *not* the same as the hp parts though. These measurements are probably not that exact. I made a really quick test setup (used a uCurrent, go eevblog!), and I'm not sure if I actually kept Vds the same between when measuring the hp parts and measuring the J310s. The hp parts do seem to be matched for Vgs vs Id, although not all that closely. You probably wouldn't want to just select two parts at random, but it'd be fairly easy to find a pair that matched close enough from an order of 10 of them.
Since A1A1Q1 just acts as a voltage follower for the high frequency components, I don't think using exactly the correct hp part is that important, although it might affect calibration slightly at higher frequencies. The lower frequency components are servo'd by op-amp U1 on A3, and should be unaffected. The higher frequency components should follow the input almost exactly since it's a voltage follower. The bias point is slightly different though, and that probably has the biggest effect on the circuit. I found it interesting that unlike my Tek 2230 (a 100MHz scope), there's no input protection diodes. I wonder if this is an oversight, or if it was too hard to get the performance they needed with protection diodes in circuit. The gate of Q1 seems to be biased by it's (extremely small) leakage current, which flows through R10.
Continuing with the repair...inputting a square wave on channels A and B gave the following waveforms:
[4a]

[4b]

Channel B apeared to only be passing the high frequencies, and channel A didn't look quite right either (both channels set as terminated in 50ohms, and the resistors measured good). Here's a slightly faster square wave on channels A and B, and also on my Tek 2230 to show what it's supposed to look like:
[5a]

[5b]

[5c]

For me, the problem with channel B ended up being one of the pins on the connector between the attenuator board and A3 was bent. After taking out the attenuator board, I bent the pin back into place from the other side. I also took out the channel B attenuator board and cleaned the contacts and reseated all of the socketed parts, and that fixed channel A not passing hihger frequencies intermittently. Similarly, most of the boards along the horizontal timebase shaft tend to get unseated with use from the force of turning the knob. I took all of these out, cleaned the contacts, and reseated them, and that helped with intermittent trigger issues. After a lot of electrical contact cleaner, I got my hp 1725a repaired.
This oscilloscope has a pretty unique feature for the delayed time base which I have never seen before. You can sweep part of the screen at a slower speed on the main time base, and then the rest of the screen at a faster sweep speed on the delayed time base. I thought this was pretty cool, because I've never seen anything like it before, but it doesn't actually work that well in practice unless the main and delayed time base are at a similar setting. Otherwise the delayed part is too dim to see.
Main intensified:
[6a]

Main and delayed on same screen:
[6b]

Other notes:
Going through the calibration procedure in the newer hp manual, I can get everything into adjustment, except the horizontal sweep speed and linearity adjustment is extremeley finicky trying to get it right throughout all ranges, especially for higher sweep speeds (sections 5-77 through 5-84). It's really hard to get the left/right side of the screen to match for all sweep speeds.
A high resolution copy of the service manual is available from qservice, and I'd highly recommend the qservice copy over the scan you can download for free from hp/Agilent/Keysight if you're working on the scope. The qservice schematics are much much clearer. The hp manual is newer though, if you have a newer serial number unit. Specifically, schematic sheet 11 is pretty much completely redrawn for the time interval marker control circuit. (The hp 1725a has pretty good specs for time interval measurements, and apparently those specs were really hard to meet. The parts placement and adjustment procedure is also slightly different for newer models for that board.