Looking at this again i would not be shocked if BT did not do this install. In my prior 5 years as an Openreach Engineer 10+ years ago in my late 20's I would had never wired anything like that. It is a mish mash of old and new wiring (does not appear to have a single blue or brown or yellow wire anywhere) It may not even be telephone cable which has been used (bodgers and alarm cable or worse door bell cable are not uncommon).
Assuming the wire which looks more red is actually orange that should be accompanied by a solid white as pair one. Which it does (assuming that is a white wire and not some pale weird green as it looks in parts of the image) on one of the wrap around splice connectors/terminals <big black thing> (why its spliced i do not know).
A second telephone line with a different phone number would terminate on solid Black, and solid Green. (your second appears to be black and orange or red as best as i can make out).
All these splices and joins will have an effect on a broadband connection. Even when things are wired properly and you connect to an extension wired correctly using the punch down terminals you will lose some speed, so i dread to think how much you are losing with that sheer utter mess.
I bet on the second line there is no improvement even when you connected to the test socket?????
Your best course would be call BT in to fix it and do things properly, though this is going to cost you money. Probably at a cost of around £120 per line. For the Master move and rewire + any additional cost for new external extension wiring (which im guessing would also be required).
The second route which i should not, do not and HAVE NOT recommended if you want to do things yourself is to buy
2x NTE5 Master sockets
2x ADSL/VDSL mk2 or 3 master faceplates Like...
http://www.run-it-direct.co.uk/btvdslfaceplate.html
BEFORE YOU READ ANY FURTHER be aware tampering with the incoming pairs and anything before the master socket is technically illegal.
Refer to prior link given
http://www.rob-r.co.uk/other/UKphonecatwiring.htm for where wires go and extensions go......
Connect the incoming pairs to each of the 2 NTE5 sockets A and B terminals on the rear of the NTE5s see they are both in a single location preferable side by side 6-12 inches apart ( this will get rid of all those terminals/crimps/connectors and huge mass of awkward wiring behind there or as you put it "There were so many wires it was difficult to put the master plate back" im not shocked with all the stupid connectors and mass of wiring).
Connect phones which need to go elsewhere to the IDC extension terminals on the back (the terminals like what is in your first image, those are what should be used)
Connect any XDSL extensions to the 2 IDC terminals on the front of the DSL faceplate as shown in the runit link provided (clearly circled in it) above.
This is what any BT engineer of any worth will do to your set up.
The set up you currently have is an utter mess no new modem is going to perform miracles with things as they are.
Sort the wiring first, or you can ignore the advice buy a new modem and fiddle for ages get maybe a small benefit but nowhere near what you could.
I in NO WAY recommend you do this job yourself especially if you are not familiar with telephone cabling. I suggest you contact BT for an idea on costs etc.