You might be interested in this kind of product - I'm not recommending a specific example, but this, or its higher-powered twin (see panel on right) might get you out of trouble.
That said, it isn't a cure for your fundamental problem - that you are going to the car, it should start and it doesn't, it is just a means to make the effects of that less severe.
Several possibilities come to mind; your new battery may not ever have got fully charged (you don't mention distances, but it takes a lot more than a five minute run to get the stupid things to fully charged), there may be something discharging the battery (what happens if you disconnect the battery overnight, for example?) or the charging may be faulty (honestly, I wouldn't really trust someone who just said 'that's ok' when you are in this situation; clearly something is wrong, and some voltage readings would be more than helpful).
There are a number of things that you could, helpfully, do, but I get the impression that you don't have a DVM or any expertise with one, or you would have said something about that and actual voltage numbers (that's just a guess, obviously).
In any case, putting the battery on charge, overnight, could help you be sure that the battery has got a decent amount of charge in it (preferably accompanied with voltage readings).
I don't suppose that you have a code reader? Something informative could show up in the vehicle fault codes.
Practically, if you can't solve this yourself in relatively short order, a visit to an Auto Electrician has got to be on the cards; it will hurt, but more bad advice from, say, an RAC man could leave you without reliable transport for far longer and just changing things, at random. (They should be able to test battery capacity and starting current, slightly indirectly, and you could then be assured (or not) about the new battery. Also, they are more likely to do a competent job assessing your alternator than a visiting get-you-going man.)
I think that checking terminal integrity and freedom from corrosion is a 'must do'; you wouldn't want to spend money on replacing stuff, if a clean and grease of the terminals is all that is required. It takes remarkably little corrosion to influence charging and the ability to deliver starting current, so this needs to be done carefully.
Halfords may have one of those starter/compressor type products at half price, but their site is currently offline. Worth having a look, I suppose.