As a first step in my planning for the trip, I looked into hiring a bicycle in Uppsala, and came across this: a bike hire company that will kit you out for a ride (with trailer) to buy up big at Ikea. Here's the pitch:
"Going to IKEA without car? Riding with four grocery bags or a bed as cargo? It's easier than you may think to transport cargo by bike, and quite practical. See the bike trailers page."
I hope the company won't mind me pasting in the demonstration photo:

You can't fault the Swedes.
(See for yourself here.)