Since they have logs of you doing it numerous times (with different proxies), the situation is pretty grim; any concept of forgiveness due to a "once-off mistake" has long gone out the window.
A lot of the answers here so far are basically speculating on whether or not you're likely to be fired, or at least seriously reprimanded. I don't live in the UK, but I think it's sufficient to summarize your situation by saying that the fate of your job really depends on:
- The disposition of your boss / senior management / security staff at the company (and whether anyone at any of those levels is willing to stick up for you);
- The letter and spirit of all agreements you've signed since hiring on at the company (including after joining, for any network or systems access you needed);
- The amount of evidence (or lack thereof) that they are able to find about the nature of your proxy accesses, what you viewed/downloaded, and what impact it has had;
- The cost-benefit analysis of: the situation- and country-specific legal burdens imposed on the company in order to justify firing you, vs. the cost of attempting to re-educate you and allow you to remain on staff, plus the risk of you doing it again and causing a more serious problem for the company;
- How well you can explain exactly what you downloaded, accessed or viewed; how contrite you are when explaining it; and the degree to which you can back up your assertions with evidence.
I'm not here to speculate about the particulars of these variables. That depends entirely on your situation, and most of these are too personal for us to reasonably evaluate from afar.
What I will offer is some advice.
As a technologist, I understand that you can very easily get frustrated by needlessly restrictive firewall blocking rules at your place of work. They tend not only to block off-task activities like Youtube and Facebook, but also useful technical journals, like the blogs of individual technologists on Blogspot, as well as, for example, GitHub Gists. You probably can't get on IRC, either, which hampers your ability to interact with open source communities.
However, in the future, you can't simply route around the damage by using an open proxy. Instead, let me propose some alternatives, in descending order of "safety"; safety for your job, as well as overall security for the network you are a steward of defending by virtue of the fact that you have access to it:
- If your IT department has a responsive way to request sites to be unblocked, and you need them for business purposes (or, honestly, if your company is easy-going, even if you want to browse harmless fun content during your lunch break), you can request that a firewall exception be made for a specific site or wildcard domain. If you aren't sure of the process to get an exception made, ask your boss, and keep escalating until you find the process. Be prepared to explain the nature of the content of the site and how you think it will benefit you (improving morale is a potential benefit even if it's a "fun" site like xkcd or something).
- Ask for an officially supported "knowledge worker" internet access system. They might already have one. For example, you might be able to remote desktop into a system that has less restrictions on the sites it can access, but that system wouldn't have any direct LAN access to valuable company assets like the file server. If they don't have such a system, ask if you can use a VPN to a server that they specifically set up to be disconnected from the network and have a "no split tunneling" policy on your VPN software; this might allow you to connect to either the company network or a less-restricted Internet pipe, but not both at the same time.
- Ask if you can connect to your own VPN. For this you will need to know how to set up a server, and pay for hosting (or host it at home if you have a sufficiently high-end connection and dynamic DNS or a static IP). Offer to let them audit the security of the system and commit that you will keep it patched, install a virus scanner/NIDS, and assume all responsibility for any breaches that trace back to your VPN usage. They might actually give you permission if you seem responsible and clueful enough.
- Ask if you can bring in your own personal smartphone (or if you are 100% sure you can, do it). Unless you work in an extremely high security environment, it's likely that your company already allows this. Look around; do other coworkers like your superiors regularly have a smartphone or tablet? If they do, just ask them if it's company property or if they own it.
- If other coworkers have their own personal cellular devices in the building and you aren't aware of this being a security violation, your best way to "view memes" (or blocked sites in general) at work is to use your phone, or get a small laptop and tether your phone to your laptop, assuming they allow you to have your personal laptop also. If personal devices are ubiquitous in your office, you shouldn't even need to ask for permission; just do it, and if you get called out, point at literally everyone else sitting around you and ask when they're going to call out the rest of your office, too.
- If other coworkers don't have their personal devices, or if personal devices are disallowed, ask around if you can get a company-owned smartphone or tablet. These are almost always "off-net" (meaning you can't access sensitive company data on the LAN), or can legitimately be configured to be so; in this case, you can access your memes on this device. If the network connection (e.g. the LTE) is provided by the company, you should be careful not to view anything that could even slightly be construed as pornography or illegal activity on the device, but otherwise, non-business access might be OK. Ask whoever issues you the device whether you have a data limit, and what happens if you reach or exceed it, and make sure to work within those bounds (if any).
- Never connect personal devices to the company network (WiFi or Ethernet) for any reason, unless you've been given explicit permission by your manager, in writing (and ideally also permission from your IT department, also in writing).
- If you are going to go "off the rails" and do something without permission, don't use an open proxy, and your proxy should not be unencrypted under any circumstances.
- Open proxies that allow just anyone to roll up and start proxying data are hideously insecure. On top of that, usually they will mine your data, monitoring both the domains you visit, as well as any traffic you produce that isn't end-to-end encrypted. If you login to an unencrypted (http) site over an open proxy, that's a very high guarantee that the proxy operator will steal your credentials. Ditto if you submit a session cookie over http.
- Unencrypted proxies are almost as bad as open proxies, even if you're legitimately renting or own the underlying hardware, because any hops along the chain that are hostile or compromised can sniff your data (basically any MITM attack would steal your data). Use a TLS tunnel (like from stunnel) as a best practice if you don't want to or can't use a proper VPN protocol.
You might not think the latter two things matter, but if you did this and got caught but were able to say, "I know I violated company policy by using an unauthorized proxy/VPN, but I used my own hardware which is kept secure and patched, and the connection is encrypted with TLS; here is a list of sites I accessed and why" -- that would significantly help alleviate their concerns (as long as those sites were generally harmless).
I'm not sure if your company has grounds to take legal action as it is, but if you had the above explanation, they might not want to press charges even if they could. Basically, all the security risks and consequences are somewhat lower if your connection is (1) not an open proxy and (2) encrypted with an industry-standard strong encryption suite.
That doesn't mean you should go off the rails even in this situation, but doing it very sparingly and for legit business purposes would probably not result in sacking (though it's always better to ask permission than forgiveness for things like this).