Welcome to Flyertalk.

Don't get your hopes up. An upgrade for a no-status passenger these days is a longshot.

IMO, you'd be better off looking for an off-season, deep discounted business class fare. Then the "upgrade" is a certainty and not "very unlikely." And sometimes the cost isn't materially different than the econ ticket fare plus fees plus taxes plus co-pay plus value (or price) of the miles. Cheap business class tickets usually require a long advance purchase and typically require a round-trip purchase and a minimum stay.

Not many no-status passengers (especially those unfamiliar with how this game works) actually have 25k or 50k miles. Buying those miles with the hopes of upgrading would be a very bad idea.

Cheap fares can sometimes be found between the USA and LHR. Sometimes you need to fly to someplace like DUB or connect from LHR to someplace nearby. Then just buy a cheap ticket for the short hop (or ride the train/chunnel depending on the location) to get back to London.

Good luck.