If you want a fully flexible ticket both ways, and don't want to use WMT, you're going to be paying an arm and a leg for an Anytime ticket.
If you're happy to commit to an Advance ticket, at least on the outward leg, it's currently £38 in standard from Lime Street to Euston on the 07:47. That's probably the latest train you want to take if you want to be there in time.
It then depends on how you want to get to Heathrow - if you take the Tube all the way, it's just £3.30 on Oyster/contactless. If you take the Tube to Paddington and then TfL Rail, it's £10.40.
If you want to take the Heathrow Express, you're best off booking in advance - if you buy "Fixed Day" tickets now, it costs £18 for the outward journey and £16.50 for the return. That would be on top of the £2.40 for the Tube to get to Paddington.
On your way back to Liverpool you are probably best off waiting until the day to buy your ticket, as obviously your flight may be delayed or in fact you may face a lengthy queue at passport control (unless it's an intra-CTA flight).
There's no advantage to buying an Off-Peak Single now, as the old "Saver Half" has been scrapped. A Single is now 70% of the Off-Peak Return regardless of how or when you buy it. And of course there may be an Advance ticket on the day that's cheaper.
As for Standard Premium, that would cost you £25 each way and is simply payable to the guard on the day, like Weekend First used to be.