In reply to gammarus:
have done similar several times.
It's doable within a single day, but it's a long day. Carry a large water bottle. 2 stops on RER line D gets you from Paris Nord to Gare de Lyon, you want about 1 hr between eurostar arrival to TGV departure times - (esp if you've not done it before) you could do it in less time but it can take you 15 min just to get off the eurostar and walk to the end of the platform. Buy Paris metro tickets from the eurostar buffet-car. All TGVs require seat booking before departure.
I put our wheely double ski bag into the overhead rack on train, UK, eurostar and TGV - it helps to get on prompt to grab the space. Do not ever travel the UK leg with lumo, they have a no sports equipment / oversize bags policy. It's been done in other forums and chats - but you absolutely can take ice-axes onto the eurostar. Going via Lille can save the pain of crossing Paris. Rail passes are well worth looking into.
this guy has almost everything you need to know: https://www.seat61.com/
Post edited at 20:31