I have some similar experience over the summer that might be informative, though the circumstances were different (illness mid journey rather than immigration issues), I suspect similar mechanisms may have come in to play in your case.

You can read the longer version here (https://www.flyertalk.com/forum/35467094-post15.html), but in brief, I arrived at Heathrow, became ill, asked to be offloaded from my longhaul flight and be returned to Glasgow.

It transpired that in order to do this, the final coupon on my ticket (LHR-GLA) was taken and used out of sequence to return me home. Despite this, the long-haul legs to and from DFW continued to display in the BA App, with seat selections, right up until the departure times, which sounds similar to what you describe above with your seat assignment. OLCI for those flights didn’t work either (I tried out of curiosity, despite being back home in Scotland).

Unfortunately for me, because the coupons were used out of sequence, the entire ticket was invalid and BA were unwilling to entertain any discussions of refunds or rebooking in future.

Something similar may have happened to you, so I’d recommend contacting BA beforehand and not relying on trying to solve at the airport.