No problem on the delayed replies
Caution: essay below...
Based on those 4G CellIDs - they're different, but it's the same mast, just different sectors - it looks like your connected with 4G to this mast to your South-East on Mode Wheel Road:
https://goo.gl/maps/1LUXQyfP5miMSuqW7.
(Note: your 3G connection may be coming from somewhere else, unfortunately there is no easy way of telling this)
However, unfortunately, there are a number of factors that are against you:
1. looking at the line-of-sight path between your postcode and the mast, there is this large set of flats in the way, which will be blocking a lot of the signal
https://goo.gl/maps/jmHcfFGWHanr5Rri8
Also, I don't know which part of the flat block you live in, or which way your windows face, but if you're in the North-West then the entirety of your block, and neighbouring block, will also be in the signal path too.
2. It seems that you're likely to be on the boundary between two cells (you're connecting to 2 different CellIDs, both with low metrics) and the antennas on the mast on google maps aren't directed towards you, rather you lie between their aim.
3. The CellIDs you are connecting to are for Sectors 71 and 73, those are generally used by Three for Band 20 (800Mhz) - Three don't own much spectrum in Band 20 so their maximum theoretical speeds they can provide on that band is only 37.5Mbps (and that's effectively 'shared' with all users who are connected to it, so in reality the speeds are much lower, like the 3-4Mbps you are experiencing).
As for trying to improve things, I would first start by forcing your router to avoid connecting to the Band 20. This can most easily be done using and Android phone and the HuaCtrl app from the play store. Though there is also a Windows PC application LTE-H Monitor
https://www.ltehmonitor.com/en/ that can do it too.
If you force 4G only (to avoid it connecting to 3G) and then disable B20 the router will attempt to connect to other available 4G bands in the area. Ideally you'd want Band 3 (1800Mhz) as that is Three's primary 4G band which has more capacity overall (than B20), but with it being a higher frequency it will more easily be blocked by obstacles.
Three also own some 2100Mhz frequency, but is what Three use primarily for their 3G technology however in some areas some of this has been reconfigured to be 4G (Band 1). Three's B1 has less capacity than their B3 does and an even higher frequency, so obstacles pose more of an issue again. However, as you are to be able to get a 3G connection (with pretty good speeds for 3G tbh) that could mean one of two things:
#1. the 3G mast you connect to is closer and/or has a better line-of-sight (i.e. its not the mast on Mode Wheel Road where your 4G connection comes from) but is a 3G only mast and doesn't serve 4G.
#2. your 3G connection is coming from the Mode Wheel Road mast but Three maybe haven't yet reconfigured some of that to 4G yet - though I feel this is less likely of the two as I'd expect the 2100Mhz higher frequency to be being blocked by the buildings in the signal path but if 2100Mhz was coming from there for 3G, B3 4G (1800Mhz) should be possible, but it its not.
Back to band forcing with HuaCtrl/LTE-H Monirot - if you can play around with forcing only B3 and then only B1 to see what is available and noting down the CellIDs and metrics, they would be useful - these connections may come from different masts and have different signal levels - those might be the ones that end up needing to be 'targeted' by using a directional antenna for the best 4G speeds.
My only other comments right now are that the B311 router is only a CAT4 device (think mobile phone from 5+ years ago), and it is only able to connect to a single band at once. Better routers (more £) are able to connect to multiple bands at once and aggregate them with an aim to increase speeds. However, given that it appears you only seem to be able to get a B20 connection at the moment we first need to see what other bands can be forced before thinking if a different router would produce better speeds.
Finally, with any speed tests you perform, try to you use a wired (ethernet) connection to a PC as that avoids any fluctuations with Wifi, which in a built up area/flats can be problematic due to interference from other wifi in the area.