Longest serving current 737-400 with BA is G-DOCB, delivered on 16th Oct 1991, so 21 years old last week. 747-400 G-BNLE wins though, delivered on 15th Nov 1989, so almost 23 years old. The 767s aren't far behind, with G-BNWB being delivered on 8th Feb 1990.
Jet2's 733s made their first flights between Aug 86 (G-CELP) and May 98 (G-GDFK - started out with BA strangely enough), while the 757s first took to the skies between Sep 87 (G-LSAG) aand Nov 96 (G-LSAK).
For what it's worth I wouldn't hesitate to get on any of them!