OP I can't dry outside, don't have a tumble dryer or radiators so find drying washing challenging.
Things smell if they don't dry quickly enough.
Check your washing machine is clean and not smelly, if so clean it first.
Give the towels a hot wash with wash powder and no softener to get rid of the smell. Then you need to dry them thoroughly and quickly.
Start by giving them an extra spin cycle in the machine, get them out as soon as the cycle ends. Then find somewhere dry and warm to hang them where you can spread them out. A clothes airer doesn't give you enough space for bath towels.
I put a heater and a dehumidifier on in the room I am drying washing in if it is cold. On warm sunny days I open the windows.
Try rigging up a washing line indoors to give you space to hang the towels without folding them or doubling them up. If you can a line above your stairs works well.
Another option is to open your curtains and use a couple of clothes hangers with the clips for skirts to hang your towels from the curtain rail. Leave the window open so you get air flow. Internal doors are another good place to consider.
Do not try to dry washing in the bathroom, it is the dampest place in the house.
Do not do all your washing at once, only try to dry one load at a time.
Do not put things away until they are absolutely dry as that makes them smell.
For other items a hot iron helps ensure things are 100% dry before you put them away.