Often1

Updated: OP answered the one vs. two ticket question while I was posting. So, ignore that part below. To be fair, while OP is frustrated, the questions asked here are legitimate and determine what will happen. There was no snark from others and if one cannot answer the harder questions, here one simply goes away. With BA, one simply loses the claim.

Whether OP was on a single PNR is irrelevant. The proper question is whether OP was on a single ticket NCL-BOS with a connection at LHR or on two tickets, even if in the same PNR, one ticket being NCL-LHR and the second being LHR-BOS. If the former, it is likely that BA simply did not cancel the entire ticket in light of the flight cancellation and ensuing backup, by the time OP boarded at LHR. If on two tickets, OP, the no show on NCL-LHR would not affect the onward LHR-BOS ticket.

OP may need to resubmit because some of this is an EC 261/2004 claim and the rest is seeking a customer service gesture. A lot will also depend on the one vs. two ticket issue, which OP really does need to answer. As to why this claim has not been expeditiously processed, it seems that BA can do a good job once it has determined that an event is compensable. E.g., everybody on a series of flights departing a given airport on a given day between specified times. Those claims are easy because they fit a bucket subject to compensation. OP's situation is much more complex and also the claim is for items which fall outside the Regulation and thus require evaluation from a customer service perspective.

1. Here, when BA cancelled the flight, OP had the option of cancelling or rerouting and was apparently afforded the reroute option, e.g. NCL-BOS to LHR-BOS. There was no delay in arrival and the time parameters for reroutes were waived by OP's choice of self-help. So, no delay compensation.

2. The train ticket is a claim against travel insurance and will be defrayed by the refund due for the NCL-LHR segment which was cancelled. BA might choose to pay for the train ticket as a customer service gesture, but there is no legal requirement (although BA does owe OP a duty of care which would cover a meal or two along the way)