Feels like you missed the point. How much of a power envelope £11/m buys you is largely irrelevant, anyone can spec a box that will do op’s required workload in 5-6w, but it’s not in any way comparable in terms of resilience or potentially speed to what op’s £11/m is currently providing.
Potentially, but I would suggest you start by diagnosing why sharing from your PC is slow first. Obviously pulling data remotely depends on your servers connection to the router, the upload speed of your connection, the connection between your ISP and wherever the remote device is and the same on the remote side to the client.
The functionality at 5-6 watts would be very limited but the hardware should be cheap as chips though.
It depends how much data the OP might store and what functionality is needed. If the task is to just backup a pair of phones I don’t think I’d bother building a system for that alone. You can buy phone docks that that will perform backups while charging, I think even even phone cases that offers extended battery life too.