It would certainly help if you told us where you want to fly TO!!!
I'm also a bit confused in that I would have assumed that "the eve of bank holiday Monday" meant Sunday evening (in the same way that Christmas Eve is the evening before Christmas Day) but you later refer to "a bank holiday Monday evening" So I'll now assume that you do want to travel on Monday.
There are currently no flights operating directly to the UK from Tenerife North Airport (but it might be a possible departure point for a connecting route).
From Tenerife South, Ryanair fly to Birmingham, Bournemouth, Bristol, Edinburgh, Prestwick, Liverpool, Luton, Stansted & East Midlands.
Jet 2 serve Leeds/Bradford, Manchester, Newcastle, East Midlands & Belfast (which is also served by Aer LIngus).
Monarch fly to Birmingham, Gatwick, Luton & Manchester.
easyJet operate to Bristol, Gatwick & Manchester.
No budget airline offers discounted tickets at short notice. They always charge the maximum fare for that route. (Only a few 'full fare' airlines are prepared to haggle over fares immediately prior to take-off, and usually only on 'medium haul' or 'long haul' routes).
Looking at travel to any London airport, Ryanair and easyJet have no seats available. Monarch can fly you to either Gatwick or Luton but it will cost you €602.50
The only way to cut costs (if it's actually possible) might be to fly to a different airport in the UK and then take the train (or another flight, if available). I'm prepared to take a look at your possible options but only if you'll tell me where it is you actually want to get to!
Chris