AFAIK, no-one has started yet on the new intake. First new entrants will appear in the New Year so won't be on line until March ish.
As you say, the offer will be for either 737s at LGW or the Airbus at LHR. No other fleets are available to new entrants such as yourselves (SSPs - self-sponsored pilots). There's probably not a lot in it between the two fleets in practice. Obviously the 737 is a more hands on aeroplane that actually needs flying. The A320 is significantly more advanced technology, however from a pure flying point of view, it's much less involved. Plus the "powers that be" won't allow you to take out the autothrottle when you're flying manually. Try that on a 737 and you'll have an interesting time.
So in summary, if it's hands-on flying you want, choose the 737. If you want to fly out of LHR above all else, choose the A320. Lifestyle has been discussed elsewhere, although LHR will have more nightstops than LGW. Having said all that, you won't get a choice of type for your first five years and will be going where you're sent, although you can express a preference.