photo credit: Plymouth City Council: The Box
Should we believe the inscription, which says that this is someone aged 74 painted in 1596 (hence born c.1522), or that it is 'Sir John Hawkins', who was not born until 1532 and died in 1595? I guess the former.
As you say, the former. The Bodleian portrait shows him greying but still vigorous; the NMM one dated 1581 not grey at all. Both on Art UK.
This is a man older than he got to. Probably a case of Sir Wishful Thinking.
The inscription looks very plausible and correct to the period. Although the left part , the age seems it may have been tampered with. It would be good to take a closer look.The picture also looks like it has been enlarged. There is a definite line above the inscription and around the other tree sides suggesting the portrait was head and shoulders . It appears to have been more than doubled in size.. Possibly to conform with the conventional 50 x 40 size in British portraits.
This painting depicts the same likeness as the portrait said to depict J. Southcote Esquire Aged 74 and dated 1596. Anonymous Sale, Christie's January 21, 1927 (no. 112) as by De Heere 'Portrait of a Nobleman, dated 1578, in brown doublet and white lace ruff, and portrait of 'F. [sic] SOUTHCOTE ESQ., aged 74, in dark dress, with white ruff (two(' sold G. Kelly £11 11); The Society of Apothecaries of London - stolen from their premises. It is illustrated on p. 211 of Sir Roy Strong's English Icon of 1969 and attributed to Strong's
'Unknown Follower of Custodis'. This is a misnomer as there is nothing to suggest that the painter trained under Custodis. Rather this painter operated in the Southwest of England either from Plymouth or Exeter. William Brooke/Browicke is a strong candidate.
Is there any biography for Southcote? If the Apothecaries had the other version after the 1927 sale they presumably had it for some such connection, but you seem to have answered the 'who is it' question originally asked.
Personally, I wouldn't be so confident about the identification. The name 'J. Southcote Esqr.' is surely in a much later hand than the other inscriptions, perhaps over 100 years later. So unfortunately there is no guarantee of accuracy.