Dual shift levers, old bike and indexed gears
Most likely your Tiagra brifters (brakes+shifters) are 8 to 10 speed depending on their age, so the cable pull is different to a five speed cassette and probably pre-dates hyperglide.

So count the number of clicks on the right brifter, and add one.  That's the gear count you need to match.  On the left side brifter you want a double, so just one click, or two if the bike has a triple.  More on this later.

You will need a replacement back wheel, cassette, and chain.  All that needs to match the gear count of the right brifter.  The OLD of the rear wheel will have to be the same or very close to the OLD of the existing wheel.  But its a steel frame so can take a small amount of cold-setting, if you make the rear dropouts parallel again afterward.

You might need a new rear derailleur, because the cassette may have a larger large and smaller small cog than the original.

You are unlikely to need replacement chainrings assuming the wear is good.  The front derailleur mech should be okay but may need to have the cage narrowed for the thinner chain.



You can do this.

I have a 1981 Raleigh Arena with 3x9 speed Shimano 105 groupset on it.  I changed the wheels and the entire transmission.  As a result of going from 27" to 700c wheels, the brake calipers didn't reach so I had to replace them with long-reach ones.  The bike now looks like this:



EDIT Another possible issue with your front chainrings is that the cable pull on the LEFT brifter might not match the inter-chainring gap.  As a somewhat halfway solution, you can leave the left downtube shifter running the front mech while connecting the right brifter to the rear mech.  Downside here is changing chainring means a downtube friction shift, but trimming this shift is easier.  Friction shifting makes a lot of things easier.

Any questions please ask.