Second Bank Account
As far as mainstream financial institutions are concerned, the simplest course would be, as RonJohn suggested, to just open an account with another bank: either one of the major high-street banks (NatWest, Lloyds, etc.) or one of the "second-tier" banks/ex-building societies (Tesco, Santander etc.). The majority of "screw-ups" in recent years have been specific to a single bank (or banking group), so the chances are good that if your day-to-day account is affected, your back-up account won't be.
Contrary to the OP's fears (and those of some other answers), there does not appear to be any great hurdle to having a second, basic, current account in the UK. I was going to edit in some examples, but Notts90 covers this far better in their answer. Meeting extra requirements (like two or more Direct Debts, and/or a minimum deposit every month) may get you benefits such as interest on credit balances or cash-back on spending, but are not a requirement for many basic accounts.
Emergency-Use Credit Card
The processing networks for credit cards (Mastercard/Visa) are substantially different from normal UK banking (BACS/Faster Payments) so glitches that affect one bank, or even the much rarer glitches that have affected the underlying banking infrastructure will probably not affect credit-card payments.
The obvious advantage of a credit-card is that you do not have to "tie up" any money "just in case". You can spend (up to your credit limit) at time of crisis and repay the balance once things have returned to normal. Even if you have to repay in a couple of installments (and thus incur some interest charges) or need to make a cash withdrawal (incurring appropriate fees), this may be an "acceptable cost".
As asgallant suggested in a comment, an alternative to a normal credit-card would be a Pre-Paid Credit Card. These might be useful if you either have a poor credit-rating or – as the OP mentions in a comment – if you have not built much of a credit-rating. The down-sides of a pre-paid credit-card (compared to a normal one) are that you have to "tie-up" the money in advance, and you may have to watch out for fees (see "How much do prepaid cards cost?" in this Money Saving Expert article, particularly the mention of Inactivity Fees).
Some other more "off-the-wall" ideas that you might pursue (after fully evaluating the risks):
While PayPal is most often used to pay merchants, or to transfer money to others, it is possible to add money from a linked bank account to your PayPal wallet (see PayPal Help: How Do I add Money to my PayPal Balance). You could thus pre-load your account with however much money you think you'll need to ride out any bank glitch. You can obviously use the money for online purchases, and while you cannot normally access it through an ATM1, you could potentially send part of the balance to a friend/relative who does have ready-cash.
Note, that like Gift Cards (below), any balance held in a PayPal account is NOT protected by the Financial Services Compensation Scheme (FSCS). Were PayPal (or a gift-card issuer) to go bust, you would be at the back of the line of creditors and be unlikely to get much, if any, money back.
1You can get a PayPal Business Debit Mastercard which – from what I can see – acts as a "normal" Mastercard, backed by your PayPal account. This does allow withdrawal of a PayPal balance from ATMs (with a flat £1/withdrawal fee), but is only available with "business" PayPal accounts in the US and UK (but the card can be used worldwide).
Amazon (or other) Gift Card
This would not be a viable solution for many people, and certainly not for large amounts of money.
Amazon (and, undoubtedly, other major retailers) allow you to load a gift-card on to your account. While Amazon doesn't (yet) sell everything, it may be enough to ride-out the occasional glitch with your main bank. Probably the main drawback is that according to their Terms and Conditions, "Gift Cards cannot be used to purchase other gift cards", so you could not send some of the balance to another person (e.g. as a way of getting cash). However, if you have a credit card attached to your account, you probably would be able to purchase a gift-card on that, that you could send to a friend.
As both Owain and Notts90 rightly point out in comments, a gift-card carries the risk of the issuer going bust and not being able to honour it (as happened to a somewhat-similar "Christmas Club" run by Farepak in 2006, and to a number of other UK high-street retailers over the past few years such as Woolworths, BHS and Toys-R-Us: see this Centre for Retail Research list for many others who have been in trouble).
If you were to pursue this idea, it would probably be best to follow the suggestion Chris H gave in a comment of limiting exposure to cover essentials and diversifying across a number of major supermarkets.