LNAT have stated:
Will special circumstances, such as illness, be considered in evaluating my LNAT result?
If you have any examination access requirements related to disability or illness, these must be notified at the time of registration, so that suitable provision can be made. If you did not tell us about your examination access requirements (so that provision was not made), or if you merely sat the test in adverse circumstances (e.g. you were unwell or distressed), then any allowance in the interpretation of your LNAT result is between you and the universities to which you have applied. Your result will not be adjusted by the LNAT Consortium or Pearson VUE.
Of course this means that either you or your reference should contact the universities you’ve applied to directly and ask if there’s an extenuating circumstances form to fill in.
Now once you’ve done that (which you’ve stated that you have), how much does that truly make a difference to your application? Unfortunately, that’s something no one can really answer for you for a competitive course. Every place has a different procedure for extenuating circumstances so contacting the universities directly may be of help to you.
I had extenuating circumstances during my UCAT testing time period and I’ve gotten offers so far. In my opinion, if you’ve done well in your grades/had a good personal statement, the way they’d probably see it is if you’ve had a similar performance to someone else then they’ll compare LNAT scores - taking your extenuating circumstances into consideration. Of course I’m comparing this to medicine but I can only imagine it would be similar.