I believe I know most of the travel card options in the UK, and I believe your best bet would be either the Halifax or Lloyds World Elite cards. I've never had one of them; they are quite new on the market, but it does sound like for £180 per year the main cardholder and supplementary cardholder can both get in via Priority Pass for free. (The Santander equivalent is main cardholder only).
I'm quite sure the HSBC Premier World Elite mentioned in the post above is the next most economical option for two persons at £195 + £60 for supplementary. There are also unorthodox ways to qualify for Premier other than the most published methods.
(I can also think of an unorthodox method you could use with American Express which would get you completely free lounge access, but it's a fair bit of hassle on a regular basis, and would rely on you both having excellent credit ratings all of the time.)
Overall, the best option would be to find somebody you know well who could give you a supplmentary card on their American Express Platinum or SBS Platinum card; that would cost £285, and get you both in for unlimited visits plus £15 credit per person at The Grain Store restaurant in LGW. Would get you in to Priority Pass lounges, Plaza Premium, Centurion, and possibly Lufthansa (depending on your ticket). Those other lounges are generally better than Priority Pass, although Plaza Premium in LHR has been going downhill a bit recently. If you don't know such a person though, then it's not an option.
If none of the above can be made to work, then last resort really would be to get a Santander World elite each, for a total of £360 I think.
NB: I don't touch Natwest or Barclays with your bargepole let alone mine! Despicable organisations both in my personal opinion. I am vaguely aware that both have had lounge access options through cards/premier style accounts in recent times, but I don't know the details or if they're still going.