Hi Alan,
Firstly, if your knowledge is patchy then in the interests of staying alive it's best to think more than twice about repairing things yourself, especially where mains wiring (e.g. to a transformer) is involved. It's probably best to take it to someone with adequate knowledge. Having said that, and also that no responsibility is accepted for anything you do as a result of my input, here are a few general comments for you (I don't have any experience with this particular amp).
You say you believe it needs a new transformer, but if so then that should be very clear - e.g. by visible damage (burning etc), by making resistance measurements of the primary & secondary windings (with no power applied) and/or by making voltage measurements at the input and output of the transformer - possibly after disconnecting the secondaries from the rest of the amp. Transformers are expensive and fail quite rarely - especially if there is no other major fault in the amp that has caused it to fail - so you wouldn't want to be replacing it unnecessarily.
The schematic is very helpful - but it does show the US version, i.e. 120V mains. If you have a version with a 120V transformer fitted and it has been connected to UK mains (now nominally 230V - but is often between 240 and 250 in practice!) then it's likely that the amp has been seriously damaged - not just the transformer.
For a solid state amp, the VA rating of the transformer usually needs to be no more than about 1.7 times the rated audio power output of the amp. So, for example, if this is a 25W amp (I don't know) then a 40 or 50VA transformer would probably be fine, though you could go to a 60 or 75 if it was important to you to get the maximum power out of the amp. Scale these figures up, depending on the actual audio power rating of this amp. Another way to judge the transformer rating is if there is any info provided on the rating plate, usually on the back of the amp. Sometimes there is a mains input power, VA, or current rating given there. The mains fuse rating indicated (for a specific supply voltage) gives only a very rough idea, but the primary current rating of the transformer (which is roughly its VA rating divided by primary voltage) should be at least as large as the fuse value [Edit: not really so - see my later post]. The schematic seems to say both 1A and 3A next to the mains fuse, but the resolution isn't adequate to be sure what it really says. [Edit: again, see my later post.]
No, the secondary voltage does not need to be more than the DC rail voltage because of rectifier losses, as you supposed. In fact, you get an apparent 'gain' in voltage from the rectification process, because the reservoir capacitors charge up to (nearly) the peak voltage of the secondary's AC waveform, not its RMS voltage (which is the one that's quoted in the transformer spec). So, for 36V DC supplies, you need approx 25V secondaries (24V would be fine). As you say, two of those secondaries would be connected in series (observing correct polarity), with the junction between them connecting to what currently goes to the centre-tap of the transformer shown on the schematic (i.e. the terminal labelled something like "G2", which connects to earth/chassis).
Finally, note that, when replacing a transformer with a completely different type to the original, there can sometimes be issues with magnetically-induced hum. This is more likely when the new transformer is mounted with a different physical orientation to the original (e.g. replacing a conventional one with a toroidal). If the original transformer has magnetic screening or has a copper flux band around it, this can be an indication that the amp manufacturer found that a transformer with low flux leakage was needed to avoid hum issues.
I hope this helps you. Please take care.
Mark.