Help please - Chain slipping badly when applying pressure

I assume the chain was on the small chainring at the front?
Hard to be sure without a side-on shot as well, but it looks like the chain is too long. You mention the problem is only on small/small combinations - this is when the derailleur has to flex right back to take maximum slack out of the chain.
You still have 4 smaller cogs to shift to but the derailleur is already pretty much horizontal without much capacity left. As mentioned, it should look like this when on small/small.
The derailleur also seems a too far rearwards. You are only on the middle cog yet it seems to be feeding onto the rear of the cassette.

I think the derailleur hanger looks bent inwards too. That would explain different shifting on the two chain rings as well.

It is very difficult to eyeball, and the photo might be misleading. Do both the jockey wheels align exactly below the sprockets on the cassette? It is a five minute job at the bike shop to check it. 10 speed is very sensitive to hanger alignment.

I have two kids and race cyclocross, so I am well used to the woes of bent derailleur hangers!

Can't say for sure without photo from the side, but the derailleur looks to be in a very strange position and fully relaxed.

Did you not shorten the chain correctly? Smaller chainring?

No tension means the chain *will* slip on bumps... even small ones.

Have you thought of getting an allotment / taking up vegetable gardening instead ?
Lots of fresh air / good company and no slipping chains.

Looks to me like your chain is too long? Derailleur looks too horizontal - it should be in this position when you are in the highest gear (small cog). It shouldn't slip if you have a new chain and cassette. I blame chain length or rear derailleur set-up.

Something looks a little wonky to me, maybe the derailleur has taken a knock and the gear hanger is slightly bent?

Then again, it could just be a trick of the light or I might have my eyes in wonky.

I wouldn't rely on old cassettes when troubleshooting - use a new chain and new cassette only, swap it between wheels.

Why not get the mech hanger alignment checked at LBS? They did that with mine when I had rear shifting woes recently, it's a 2 minute job.

Did you remember to put the 10 speed spacer behind the cassette when fitting the new one?

They don't appear to Mavic or 11 speed hubs from what I can see in the picture, but if they did happen to be, you would also need the Mavic specific or 11 speed specific spacer too.

The chainring looks to be okay condition, but have you taken them off recently? Those Shimano F type rings have an offset, so have to fitted the correct way round.

Jockey wheels should last years and work fine, even when worn.

A chainring would have to be very badly worn for a chain to jump or skip. It won't be the same as a cassette-chain mismatch (and the latter is far more likely).

Are you sure it's only on the inner chainring?

If you're sure you have replaced both chain and cassette then I'd look at the gear cable or the rear derailleur.

If the chain is slipping up and over the cassette teeth or the chainring teeth (i.e. not changing gear side-to-side) then the chain is not sitting in the teeth properly.

Interesting that you say it occurs on the smallest cogs. These require the chain sideplates to sit in the gap between the cogs so that the rollers engage the teeth. A chain that was too wide won't fit into the gap but you have checked that?

A skipping new chain normally indicates that the cassette teeth are worn.
If you also have a new cassette then look at the distance between the top jockey wheel on the derailleur and the cassette to make sure the chain is being fed onto the teeth properly with only a minimal gap between the jockey wheel and the cassette (look up the "B-screw" on t'interweb) and that the top jockey is not behind the wheel axle (feeding onto the back of the cassette so it only catches 2 or 3 teeth).
Sometime the rear derailleur doesn't pivot (rotate) fore/aft and becomes stuck - this will also lead to poor chain feeding.

I'd suggest rechecking the chain length first, try putting a bit more tension into the derailleur to close it around the cassette, and failing these replace the chain ring.

Have you notice whether it's front or rear slipping?

Another vote here for potential cable issue- I had this, the chain slipping was in fact the mech moving around and trying to half change gear. I think this happened during big efforts as the rear end flexed enough to cause some cable movement. Shifting was also a bit of a pain at this point (but mostly OK), new cables made a huge difference .

It's also worth checking the gear cable. If there is a kink in it, or if the outer is damaged or not properly seated somewhere, then that can act a bit like a spring and cause similar behavior.

I had this problem on new bike when I swapped the cassette out for a new 11-28 (from 11-25) and replaced the chain at the same time - I figured I'd need a slightly longer chain. Despite careful measurement the chain was too long, it had too much slack and slipped - particularly when under pressure, changing gears or over rough bits of road. I just shortened the chain of course, worked fine ever since.

Check that you tightened the lockiring fully on the new cassette and there is no play in the cassette on the free hub..
I do find when I change my chains and cassettes that sometimes (and I've never worked out why!) it takes a few rides to settle in, as initially I do experience chain skip..

Silly question, but you've bought the right width chain? Could cause the slipping issue. Road chains are generally 3/32", 11 speed is slightly narrower than 8/9/10 speed. single speed/fixed is different altogether at 1/8".

i would look to the cassette, very unlikely to be the chainring. The attached was what was left of a Campag Record Chainring when it started to slip.

Jockey wheel is likely to show as worse shifting in my experience.

Is it the same on all front rings? If not then you need to replace the one that's worn (just had exactly the same - replaced cassette & chain - slipfest on one ring - ring replaced, happy days)