I would use a transformer with 2x24 secundaries (it is cheaper than a insulation transformer). Wire the secundaries in series: the 48V AC are fine for the UCL82. A voltage triplier will get the 150V for the anodes. The UY85 tube will not be reused. You only need to buy 1x 24+24V transformer, 3x 47uF 350-400V capacitors (100uF would be best - you will find one capacitor of this kind inside any scrapped consumer electronics power supply) and 3x UF4007 diodes. There is a even cheaper option: use a 2x48V or 2x55V transformer. They are bulky and may be comparatively costly when bought new (the best RS components price today is 15 euro+VAT+shipping) but are a common surplus item. Connect the secundaries in series and use the center tap for UCL82 the filament. Use a UF4007 diode to get a 120-130V supply for the anodes. I suggest to use this diode, the modern equivalent of the UY85 tube, because single wave rectification would result in 50Hz hum that is better masked by the small speaker and transformer compared to the 100Hz hum you would have with a the today ubiquitous full wave bridge rectifier. 120V is lower than the optimal value but the circuit will still work. If you plan to buy new components, I strongly suggest to buy a replacement for C9 (100nF 630V is fine) and add a bigger filter capacitor in parallel to C2 (220uF 200V, as example). Non Hi-Fi performance with audible hum and limited frequency response are to be expected, but despite this (on my experience) those cheap record players with UCL82 had a better sound than the average modern small computer/bluetooth speaker made of plastic.
You have done the right thing by dismantling the original amplifier. It had major electrical safety flaws - the lack of the insulation transformer is only one of them. I attach the original schematic for hystoric purposes only, it should not be used to build an amplifier today. Some capacitor/resistor values are tuned to compensate the original loudspeaker lack of high frequencies and the 50Hz hum.