The J14 gateway is a RAK833 on a Pi setup which uses a small piece of wet string as a starter antenna: https://uk.pi-supply.com/products/ipex-ufl-coil-spring-antenna-for-rak833 - and they have to remember to wet the string when it dries out.
These are absolutely fine for indoor work but, in context, my house setup is the same kit, a RAK833 on Pi with that antenna which is in the front room and nothing I do in my office with a test node can be heard by it. The office is at the end of the garden, so we are talking 15m, two brick walls & two sorts of French windows, not exactly a huge distance but these antennas aren’t really designed for gateways.
I very much doubt the location is exactly J14 as the unit isn’t an outdoor gateway. So you may find that you have to track down the property it is in and park outside to get a result.
Whereas the gateway at the south end of MK is an outdoor model with an actual antenna that can only feasibly be mounted outside as it is at 5m (should say 97m as it’s meant to be altitude above sea level). Given the proximity of Bletchley, if I was a betting man, I’d posit that this is a radio amateur’s setup - if they have a 1m glass fibre antenna AND they have multiple other antennas, you may be able to ask for help - standing 2m apart, naturally. Or go to Bletchley to the RSGB museum and ask for help.
In summary, I’d drive south and if you’ve got the setup done as per documentation for the Pi HAT and TTN, OTAA will be fine - I just use ABP as a sledge hammer route as it doesn’t have to mess about with all the join handshaking that you’ve observed.
At the very worst, I can sell you a RAK7246 gateway