TDS Credit

If payment for professional fee was made on 15/01/2020, that liability to deduct tax was arose on that date only. If deducted on 15th May and paid on 7th June, than it's late Deduction by deductor.
Deductor have to show that in Q4 F.Y 2019-20 return and same is reflected in 26AS of F.Y 2019-20, and credit is taken by deductee in F.Y 2019-20 itself.

What if he paid on quarter 1 on next year when will it relect in 26as and when will deductee claim?

If the professional fee paid on Q1 of 2020-21, but if fee is related to service given in 2019-20 than Deductor have to show that in F.Y 2019-20 return and same is reflected in 26AS of F.Y 2019-20, and credit is taken by deductee in F.Y 2019-20 itself.

If the professional fee paid on Q1 of 2020-21, is related to service given in 2020-21 than Deductor have to show that in F.Y 2020-21 return and same is reflected in 26AS of F.Y 2020-21, and credit is taken by deductee in F.Y 2020-21 itself.

criterion is that , you have to take year whichever is earlier (i.e Date of payment or complation of services Whichever is earlier)