There are other factors than just a simple speed test for a few seconds that determine constant streaming. You could have packet loss, which you would barely notice on a quick speed test, but would notice on constant streaming. Youtube is a terrible comparison since it's probably the most advanced streaming available (changes quality based on latency and adjusts as needed, not to much if needed it buffers much more than 3-4 seconds).
Now, since you've already replaced the desktop, it's safe to say it wasn't the computer.
Your ISP is going to tell you everything checks out if they get a signal from your modem, so they aren't going to be any help.
Based on what you've described, I'd have to say your cause is the modem/router. Depends on your setup. You either have a modem and a router, or a combo deal modem/router. If you have a standard router, replace the router. If you have a combo deal that you lease from your ISP, request that they replace it.
One test you can try (not full proof), is to power cycle the modem and/or router by unplugged it from the wall. Give it a minute, plug it back in. Wait another minute, and then test your iplayer again to see if streaming has improved.
There are other factors as well. Wi-fi for example is slower (especially with a moderately weak signal), if you're using a wireless card, try (as a test) straight plug ethernet.
If you have other devices on the network, they share bandwidth, regardless of what your speed test shows.
If you're curious about your packet loss go here:http://www.pingtest.net/index.php
Test it against BBC's location, somewhere in the UK if I recall, they should have an option for it (any hop will be fine).
If the test doesn't start automatically, click your results at the top, and click start test in the middle. Then choose your location. You should get 3 results, ms, jitter, and packet loss.
That should get you started, if you want me to look over your rest results just post them back to me.