Mere speculation. While the LDs may gain some seats at the next general election (after all they are starting from a low base of 11 won at the last general election), the idea that there might be a Labour-LD coalition is somewhat unlikely for ideological reasons. The laissez faire liberal approach on social and economic policy advocated by the LDs is very different from the authoritarian state-run socialism of the Labour party. A minority Labour administration is more likely if the Tories can't form a government.
Anything we say is mere speculation (unless we have a way of rigging elections), the point being that under normal circumstances a minority Labour administration would be a likely outcome.
However given the level of infighting, cost of living increases and various other factors, some of the more out there outcomes could come about.
It has to be remembered that in a lot of the South, where the Tories have some fairly safe seats, that Labour often sit in third place and so would unlikely to win such seats.
For instance Hampshire has 18 MP's, the chances of more than 4 being Labour is fairly low, however it could be that (if the electorate decide to punish the Tories) a few others go to the Lib Dems. Likewise Surrey, with its 11 MP's, is unlikely to see many areas vote in Labour, so again there could be a few more Lib Dems. Oxfordshire, Dorset, Kent and Buckinghamshire (maybe even Somerset, Devon and Cornwall) would probably follow similar patterns where there's likely to be limited scope for Labour to win many seats, but if the Tories get voted out in those areas it's likely to go Lib Dems.
If you look at the number of seats in England that Labour won in 1997 it was 328, as such a Labour total of about 325 (OK then there were 34 Welsh Labour seats and there could still be a few from Scotland in any future election) isn't a crazy outcome, especially given that in 2017 328 was what Corybn won. It also has to be remembered that they currently have about 200 MP's so to get to 325 would be a fairly big improvement in what they've had of late.
Yes the Lib Dems getting about 100 seats would be fairly unlikely, but not impossible. Getting 62 could be doable.
To give an example, locally to me what used to be over 65% vote for the Tory Party at the 2015 General Election (plus 8.8% for UKIP) and 75% within my local wards for the District Election has fallen, 2019 saw the General Election vote fall to sub 60% (with 0% for UKIP, with few of those votes likely to move to anyone other than to the Tories) and then fall again to 47% (the Lib Dems got 40%) in my local ward in the District Election in 2022.
Now I'm not saying that the Tories would lose that seat, however based on that I could see that seats where the vote share wasn't as safe that it could shift to see a Lib Dem win. Where as Labour would struggle to get anywhere close enough to win.
As I said it very much depends on what happens between now and the election and what each party has as policies.