Solved - in a rather strange way...

I was contacted by The UK National Lottery who had been speaking to my ISP - they wanted to forward some instructions for IE (which I don't tend to use) to see if that helped.

Very simple really - just setup IE to point to a proxsy server.
------------------------------------------------------------ 
In Internet Explorer, select 'Tools', then 'Internet Options', then 'Connections' and finally 'LAN settings'. 

Please make sure 'Use a proxy server for your LAN' is checked, 
In the 'Address' box, please type 'webcache.virginmedia.com' and enter '8080' for the 'Port' setting. 
Ensure 'Bypass proxy server for local addresses' is unchecked. 
Please 'OK' settings and Apply. 
------------------------------------------------------------ 


And that was it- IE was able to connect to the site.


What I wan't expecting was that Firefox and Chrome were also then able to connect.


Then I booted into Ubuntu (since IE isn't on Ubuntu for some reason) and both FireFox and Chrome were also able to connect.


The only thing I can think of was that one of Virgin Medias servers had decided that a particular route was my preferred solution, so I thought I would compare a tracert I took earlier with a fresh one - and sure enough there were some differences* in the route that my traffic took in the Virgin Media network.


So there you go - having problems connecting to The national lottery website - set up your proxy server as detailed above.


Simple really I guess, though of course there was no way of knowing this until I kicked up a fuss and someone looked at the issue. 

*if you care:
Here is the route when I was being refused access
  2     7 ms     5 ms     5 ms watf-core-1a-ae1-2985.network.virginmedia.net [80.3.1.37] 
  3     6 ms     6 ms     7 ms brnt-bb-1a-ge-320-0.network.virginmedia.net[212.43.162.138] 
  4     9 ms     9 ms    11 ms nrth-bb-1b-as4-0.network.virginmedia.net[212.43.162.218] 
  5    11 ms    72 ms    10 ms tele-ic-4-ae0-0.network.virginmedia.net[62.253.174.18] 
  6    10 ms    11 ms    11 ms  ldn-b4-link.telia.net [213.248.70.29] 
  7   136 ms    28 ms    10 ms  ldn-bb1-link.telia.net [80.91.246.144] 
  8    87 ms    85 ms    85 ms  ash-bb1-link.telia.net [80.91.251.207] 
  9   101 ms   102 ms   100 ms  atl-bb1-link.telia.net [80.91.247.173] 
10   115 ms   113 ms   113 ms  mai-b1-link.telia.net [80.91.252.58] 
11   115 ms   115 ms   115 ms asurnet-ic-134141-mai-b1.c.telia.net[213.248.72.6] 
12   120 ms   118 ms   121 ms tengige2-1.usa.nmi-edge05.columbus-networks.com [63.245.5.85] 
13   132 ms   131 ms   159 ms  63.245.69.190 
14     *        *        *     Request timed out. 
15   126 ms   125 ms   123 ms  24.244.158.2 
16   133 ms   142 ms   142 ms  208-87-34-15.securehost.com [208.87.34.15] 




Here is a  traceroute when working config: 
  2     7 ms     7 ms    16 ms watf-core-1a-ae1-2986.network.virginmedia.net [80.3.1.41] 
  3     7 ms     6 ms    14 ms brnt-bb-1a-ge-320-0.network.virginmedia.net[212.43.162.138] 
  4    63 ms    61 ms    61 ms  so-10-3-0-xcr1.lnd.cw.net [195.2.9.149] 
  5   135 ms   167 ms   131 ms  xe-5-3-0-xcr1.nyk.cw.net [195.2.25.197] 
  6   140 ms   144 ms   141 ms  xe-0-0-0-xcr1.ash.cw.net [195.2.21.182] 
  7   103 ms    94 ms    95 ms  equinix-ash.epoch.net [206.223.115.52] 
  8   167 ms   167 ms   168 ms  ae3-10.c01.ash.bb.megapath.net[155.229.57.180] 
  9    96 ms    95 ms    95 ms ae8-0.asbnvacz-mxc2.bb.megapath.net[155.229.101.149] 
10   167 ms   167 ms   167 ms ae3-0.lsancagb-mxc2.bb.megapath.net[155.229.57.113] 
11   171 ms   167 ms   167 ms ge-0-1-0-0.c00.lax.bb.megapath.net[155.229.120.21] 
12   178 ms   171 ms   167 ms  gig1-1.e40.lax.bb.megapath.net[155.229.120.98] 
13   176 ms     *      175 ms 209-101-27-70.lax-e100.cust.gw.epoch.net[209.101.27.70] 
14   176 ms   169 ms   165 ms  whatismyipaddress.com [66.80.82.69]