Preamble...
I can confirm the low speed (of single threads) issue is absolutely not related to IP address.
Having noted my current speed (of an 80Mbps connection), again for single thread downloads, is ridiculously low (TBB speedtester comes out at around 3.5Mbps - but it generally gives me about half the result from the BT and other speedtesters - TBB test download files are also around 3.5Mbps), I tried your trick (even though I said I wouldn't - I was too interested to see if it worked) and rebooted the HH3.
I now have an IP address in the 86 range (instead of the 31 range) and download speeds are exactly the same (slower than my old 4.5Mbps ADSL2+ connection with BE).
BT speedtester currently giving a result of 8Mbps from my (no surprises) maximum IP profile of 77.43Mbps.
Speedtest.net is about the same.
Zero point doing the "last test required" on the BT tester as the only reason it's under performing is THEY are throttling.
BT's throttling is particularly bad this morning.
Even twenty threads via Usenet is not maxing out my line.
So, in order to prove the total throughput of my 77.43Mbps IP profile is where it should be (around 75Mbps - give or take) I now have twenty Usenet threads and twelve download manager threads going at the same time.
Guess what?... My total useable throughput is 74.6Mbps (according to TBBmeter).
This is made up of 33.3Mbps from the 12 download manager threads (downloading a TBB test file) - so just under 2Mbps per thread, and around 41Mbps from Usenet (just over 2Mbps per thread).
With the TBB download test completed, Usenet total has gone up to around 55Mbps (2.7Mbps per thread).
There honestly doesn't seem to be any point in ringing India (there's no way this will be on their script, so they'll probably tell me I need a new Home Hub, or something equally stupid).
There's also very little point BT_care (on here) getting involved, as I highly doubt they've got the authority to purchase additional transit capacity (be it on a national, exchange, or cabinet level), nor do I suspect they have the authority to switch off the throttling.
This really is down to BT (they need to spend some more money and buy sufficient capacity to actually carry the double speed they're rolling out).
Actual point...
I think your problem may well be the HH3 (especially if the download speed is down at 0.5Mbps, whilst the IP profile remains unchanged).
A reboot/reconnect may well kick-start the HH3. The fact you get a new IP [at the same time] is probably irrelevant. I suspect it's the act of rebooting the HH, rather than the reallocation of a new IP, which is temporarily fixing your problem.
For your problem, you may actually get a result from calling India (the need for a replacement Home Hub may well be on their script)...Good luck!
Ade
vDSL2 FTTC Infinity with BT
DL Sync 80Mbps
UL Sync 20Mbps