I can think of no sensible route not using LU, but I would simply recommend the 205 bus as someone else already has. I have only ever done it in the early hours of the morning, so I can't comment on what the scenery may be like. I have however rode my cycle between the two and it's not a bad bit of London to enjoy at street level, so the bus journey may actually be quite enjoyable.
I've just remembered something, and I recall a Geoff Marshall video about doing London Bridge to Liverpool Street on foot and it is not that far. I haven't tried it myself yet, and I also recall the timing between walking it and using the Northern line out of London Bridge was in favour of walking it.
So that could be the alternative by rail, St Pancras Low Level to London Bridge on Thameslink then walk. Which is a bit of an odd choice, when you have the S7 stock available. If the sub-surface lines aren't open, then the 205 bus automatically would win for me.
Oh, wait, another one. St Pancras International to Stratford International, DLR to Stratford and mainline train to Liverpool Street.
I've just revisited the opening post and it mentions no walking. Oh, well, never mind! I must try the London Bridge to Liverpool Street walk one day, it sounds like a nice little challenge and if the mood takes I can pop into a Pret on the way

So yes, back to topic, forget about the railway route avoiding LU and jump on the regular 205.