Before I give my main answer, as discussed in the comments, it may be worth noting another approach that can sometimes work for unblocking some sites, and that is using Google translate as a proxy. In combination with changing your DNS settings, sometimes this can be successful in unblocking some sites.
You can browse whole sites via Google translate, using an url such as this and you can select a translation setting so that the actual 'translation' will still be in English. Just change the u=http://www.bbc.co.uk in the above url to switch sites when you want to.
You can also change your DNS settings from your ISP to Google DNS, which may help if there is some DNS filtering done by your ISP, though it might not help if your ISP is using a transparent proxy to filter material. See my answer for this question for how to change your DNS in network-manager without needing to touch your router.
Obviously TOR is the service to use for anonymity, but because of the lack of speed and problems you had running it, perhaps a free vpn service is what you want, although there are very few services around.
I must say that I am wary of free proxies and vpns and am reticent to recommend any I haven't used personally, so the only one I can report that I have experience with on Ubuntu is one called SecurityKiss. I have found the service useful on the occasions that the ISP was blocking a legitimate site and changing the DNS did not solve the problem.
It must be said that the limit for free users is only 300 mb per day, which is fine for hours of surfing, but not for hours of downloading.
The service takes a while to setup, and is not exactly a 'quick' setup. You can either use PPTP or openvpn, but I usually prefer openvpn, so you need to install openvpn and the openvpn plugin for network-manager:
sudo apt-get install openvpn network-manager-openvpn
There are extensive notes at the SecurityKiss site specifically for Ubuntu, with ample screenshots, so on this occasion it is not necessary to simply repeat them all here.
However, the drawback is that before you can follow the above instructions you need to get a configuration generated for Linux based systems from here and then log on to the client area here and download your package which contains further configuration details.
The explanation detailed here on the official site is very good, and your SecurityKiss entry in Network Manager should look similar to screenshot below by the end. I have obscured the gateway in the screenshot below.

Once set up it will be available to use from network-manager.

It is worth trying the Google translate and Google DNS combination first and then the SecurityKiss solution if you need to; if neither of these are satisfactory you might have to try TOR again.